Glossary

Short definitions of the terms this book uses, from A to Z, each with a link to the chapter that teaches it.

A

Abelianisation The quotient of a group by its commutator subgroup: the largest abelian quotient. Adding the relation that all generators commute turns a presentation of a group into a presentation of its abelianisation. See Chapter 15.

Abstract simplicial complex A collection of finite sets (the simplices) closed under taking subsets. It records which vertices span simplices without placing them in space. See Chapter 16.

Affine chart One of the standard charts of real projective space \(\mathbb{RP}^n\): the points whose \(i\)-th homogeneous coordinate is nonzero, identified with \(\mathbb{R}^n\) by dividing by that coordinate. See Chapter 11.

Affinely independent Points \(v_0, \ldots, v_n\) in \(\mathbb{R}^N\) are affinely independent if the vectors \(v_1 - v_0, \ldots, v_n - v_0\) are linearly independent. Such points span an \(n\)-simplex. See Chapter 16.

Alexander horned sphere A wild embedding of the 2-sphere in \(\mathbb{R}^3\), found by J. W. Alexander in 1924, whose outside is not simply connected. It shows that the Jordan-Schoenflies theorem fails in three dimensions. See Chapter 1 and Chapter 5.

Amalgamated free product The group \(G_1 *_H G_2\) obtained from the free product of \(G_1\) and \(G_2\) by identifying the two images of a common group \(H\). It is what the Seifert-van Kampen theorem produces. See Chapter 15.

Ambient isotopy A continuous deformation of the whole of space, through homeomorphisms starting at the identity, that carries one knot to another. Knots related this way are equivalent. See Chapter 15 and Chapter 18.

Amphichiral A knot equivalent to its own mirror image, such as the figure-eight knot. A knot that is not is chiral. See Chapter 18.

Andrews-Curtis conjecture The open conjecture (1965) that every balanced presentation of the trivial group can be reduced to the trivial presentation by a specific list of moves. See Chapter 20.

Angle defect At a vertex of a polyhedron, \(360^\circ\) minus the sum of the face angles meeting there. Descartes' theorem says the defects of a convex polyhedron add up to \(720^\circ\). See Chapter 2.

Antipodal map The map \(x \mapsto -x\) of a sphere, sending each point to the opposite point. On \(S^n\) it has degree \((-1)^{n+1}\). See Chapter 14 and Chapter 17.

Anyon A particle in two dimensions whose exchange, by braiding, acts on the state by a matrix. If these matrices do not commute the anyons are non-abelian, the basis of topological quantum computing. See Chapter 20.

Apex The point of a cone \(CX\) to which the top copy \(X \times \{1\}\) is collapsed. See Chapter 6.

Arc (of a knot diagram) A piece of a knot diagram running from one undercrossing to the next. Tricolourings colour arcs, and the Wirtinger presentation of the knot group has one generator per arc. See Chapter 15 and Chapter 18.

Argument principle For a loop in the plane avoiding the roots of a polynomial, the winding number of the polynomial's values around 0 counts the roots inside the loop. It underlies the topological proof of the fundamental theorem of algebra. See Chapter 14.

Arzelà-Ascoli theorem A set of continuous functions on a compact metric space has compact closure in the uniform metric exactly when it is bounded and equicontinuous. See Chapter 8.

A-smoothing and B-smoothing The two ways to remove a crossing of a knot diagram by joining its four ends in pairs: the A-smoothing joins the A-regions, the B-smoothing the B-regions. They are the moves behind the Kauffman bracket. See Chapter 18.

Atlas A collection of charts whose domains cover a manifold. See Chapter 11.

Axiom of choice The axiom that from any family of nonempty sets one can choose one element of each. Tychonoff's theorem for infinite products is equivalent to it. See Chapter 8.

B

Barcode The output of persistent homology: one bar for each homology feature, running from the scale where it is born to the scale where it dies. Long bars are features of the data, short bars are noise. See Chapter 17.

Base point The chosen point at which every loop in a fundamental group starts and ends. In a path-connected space the fundamental group does not depend on it, up to isomorphism. See Chapter 13 and Chapter 13.

Basis (of a topology) A collection of open sets such that every open set is a union of members of it. A collection of sets that covers \(X\) and in which each point of an intersection of two members lies in a member inside that intersection is a basis for the topology it generates. See Chapter 4.

Betti number The \(n\)-th Betti number \(b_n\) is the rank of the \(n\)-th homology group: roughly, the number of independent \(n\)-dimensional holes. For a graph, \(b_1 = E - V + c\) is the cycle rank. See Chapter 16 and Chapter 3.

Binary icosahedral group The group of order 120 that double covers the rotation group of the icosahedron. It is the fundamental group of the Poincaré homology sphere, and it is perfect. See Chapter 19.

Borel conjecture The open conjecture that two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic. See Chapter 20.

Borsuk-Ulam theorem Every continuous map \(f: S^n \to \mathbb{R}^n\) sends some pair of antipodal points to the same point. In dimension 2: at any moment there are two opposite places on Earth with the same temperature and pressure. See Chapter 14 and Chapter 17.

Bottleneck distance A distance between two barcodes: the smallest \(\varepsilon\) for which the bars can be matched with endpoints moving by at most \(\varepsilon\), short bars being allowed to match nothing. See Chapter 17.

Boundary (of a chain) The image \(\partial c\) of a chain under the boundary map, which sends each oriented simplex to the alternating sum of its faces. An \(n\)-chain that is a boundary of an \((n+1)\)-chain is an \(n\)-boundary; \(\partial \partial = 0\). See Chapter 16 and Chapter 16.

Boundary (of a manifold) The points of a manifold with boundary that have a neighbourhood like a half-space, with the point on the edge, rather than like \(\mathbb{R}^n\). The boundary of an \(n\)-manifold with boundary is an \((n-1)\)-manifold. See Chapter 10 and Chapter 11.

Boundary (of a set) The points of a space that lie in the closure of a set and in the closure of its complement: \(\partial A = \overline{A} \setminus \operatorname{int} A\). See Chapter 4.

Bounded A set in the plane, or in a metric space, that fits inside some large disc or ball. See Chapter 1.

Box topology The topology on a product \(\prod X_i\) whose basis is all products \(\prod U_i\) of open sets. For infinite products it is finer than the product topology and worse behaved. See Chapter 6.

Bracket (Kauffman bracket) A Laurent polynomial \(\langle D \rangle\) in a variable \(A\) assigned to a knot diagram by smoothing every crossing. It is unchanged by Reidemeister moves R2 and R3; correcting for R1 with the writhe gives the Jones polynomial. See Chapter 18.

Braid A collection of strands running monotonically from one row of points to another, up to deformation. Braids on \(n\) strands form the braid group; braiding anyons is the operation of topological quantum computing. See Chapter 15 and Chapter 20.

Braid group on three strands The group \(\langle a, b \mid aba = bab \rangle\). It is isomorphic to the knot group of the trefoil. See Chapter 15.

Branch point A point of a shape with index 3 or more: three or more arms meet there, as at the centre of the letter T or Y. See Chapter 1.

Brouwer fixed point theorem Every continuous map from a closed ball \(D^n\) to itself has a fixed point. It is proved for \(n = 2\) with the fundamental group and for every \(n\) with homology. See Chapter 14 and Chapter 17.

C

Cantor set The set left from \([0, 1]\) by repeatedly removing open middle thirds. It is compact, totally disconnected and has no isolated points, and is homeomorphic to the product \(\{0, 1\}^{\mathbb{N}}\). See Chapter 7 and Chapter 8.

Cartesian product The set \(X \times Y\) of ordered pairs \((x, y)\) with \(x \in X\) and \(y \in Y\), and more generally \(\prod X_i\). See Chapter 6.

Casson handle An infinite tower of kinked handles that Freedman showed is homeomorphic, though not always diffeomorphic, to an ordinary 2-handle. Casson handles are the tool behind the topological 4-dimensional Poincaré conjecture. See Chapter 19.

Catenane A molecule made of interlocking rings, a link in chemistry. See Chapter 18.

Cell complex A space built by attaching discs of increasing dimension along their boundaries, starting from a set of points. Also called a CW complex. See Chapter 6 and Chapter 2.

Cell decomposition A division of a surface into vertices, edges and faces, each face a disc whose boundary runs along edges. The Euler characteristic is computed from one. See Chapter 2.

Chain map A sequence of homomorphisms \(f_n: C_n(X) \to C_n(Y)\) that commutes with the boundary maps. A chain map induces homomorphisms on homology. See Chapter 17.

Chain (simplicial) An \(n\)-chain is a formal integer combination of oriented \(n\)-simplices of a simplicial complex, added and subtracted like vectors. The \(n\)-chains form a group \(C_n\). See Chapter 16.

Chart A homeomorphism from an open set of a manifold to an open set of \(\mathbb{R}^n\): a local coordinate system. See Chapter 11.

Chern number An integer attached to a complex line bundle over a closed surface. In the quantum Hall effect it is the integer that the Hall conductance measures. See Chapter 20.

Chiral A knot that is not equivalent to its mirror image, such as the trefoil. The Jones polynomial proves the trefoil chiral. See Chapter 18.

Chromatic number The smallest number of colours needed for a proper colouring of a graph. See Chapter 3.

Classification of surfaces Every closed surface is homeomorphic to exactly one of: the sphere, a connected sum of \(g \ge 1\) tori, or a connected sum of \(k \ge 1\) projective planes. Two closed surfaces are homeomorphic exactly when they have the same Euler characteristic and orientability. See Chapter 10 and Chapter 10.

Clopen A subset that is both closed and open. A space is connected exactly when its only clopen sets are the empty set and the whole space. See Chapter 4 and Chapter 7.

Closed 3-manifold A compact, connected 3-manifold without boundary, such as the 3-sphere, the 3-torus or a lens space. See Chapter 19.

Closed map A map that sends closed sets to closed sets. A continuous map from a compact space to a Hausdorff space is closed. See Chapter 8.

Closed set A set whose complement is open. Finite unions and arbitrary intersections of closed sets are closed. See Chapter 4.

Closed surface A surface that is compact and connected and has no boundary, such as the sphere or the torus. "Closed" here means "finite with no edge", not "closed subset". See Chapter 2 and Chapter 10.

Closure The smallest closed set containing a set \(A\), written \(\overline{A}\): the intersection of all closed sets containing \(A\). A point is in the closure exactly when every open set around it meets \(A\). See Chapter 4.

Coarser topology A topology \(\mathcal{T}\) is coarser than \(\mathcal{T}'\) on the same set if \(\mathcal{T} \subseteq \mathcal{T}'\): it has fewer open sets. The other is then finer. See Chapter 4.

Cocountable topology The topology on a set whose open sets are the empty set and the sets with countable complement. On an uncountable set, sequences cannot detect its closures. See Chapter 4 and Chapter 5.

Cofinite topology The topology on a set whose open sets are the empty set and the sets with finite complement. On an infinite set it is T1, compact and not Hausdorff. See Chapter 4 and Chapter 9.

Colouring (proper) An assignment of colours to the vertices of a graph, or the regions of a map, such that adjacent ones get different colours. See Chapter 3.

Comb space The union of \([0, 1] \times \{0\}\) with the vertical segments over \(0\) and over each \(1/n\). It is contractible but does not deformation retract onto some of its points. See Chapter 12.

Compact A space is compact if every open cover has a finite subcover. Compactness is a topological invariant, and continuous images of compact spaces are compact. See Chapter 8.

Compactness theorem (of logic) If every finite subset of a set of logical sentences has a model, so does the whole set. One proof is an application of Tychonoff's theorem. See Chapter 8.

Complement The set of points not in a given set. The complement of a simple closed curve in the plane has two components. See Chapter 1.

Complete bipartite graph The graph \(K_{m,n}\) with \(m + n\) vertices in two groups and an edge between every vertex of one group and every vertex of the other. \(K_{3,3}\) is not planar. See Chapter 3.

Complete graph The graph \(K_n\) with \(n\) vertices and an edge between every pair. \(K_5\) is not planar. See Chapter 3.

Component A maximal connected subset of a space. The components of a space are disjoint, closed, and cover it; homeomorphic spaces have the same number of them. See Chapter 7.

Concatenation (of paths) The product \(f \cdot g\) of two paths with \(f(1) = g(0)\): run through \(f\), then through \(g\), each at double speed. See Chapter 13.

Cone The space \(CX = (X \times [0, 1]) / (X \times \{1\})\), a cylinder on \(X\) with its top squashed to a point. Every cone is contractible. See Chapter 6.

Configuration space The space of all positions of a mechanical system, such as a robot arm or a rigid body. Such spaces are often manifolds: a rigid body's is \(\mathbb{R}^3 \times SO(3)\). See Chapter 11.

Conjugacy problem The problem of deciding, from a presentation of a group, whether two words represent conjugate elements. For fundamental groups it asks whether two free loops are homotopic. See Chapter 15.

Connected A space is connected if it is not the union of two disjoint nonempty open sets. Intervals are the connected subsets of \(\mathbb{R}\), and continuous images of connected spaces are connected. See Chapter 7.

Connected sum (of knots) The knot \(K_1 \# K_2\) made by cutting each knot open and joining the ends side by side. See Chapter 18.

Connected sum (of surfaces or manifolds) The space \(M \# N\) made by removing a small open disc from each and gluing the two boundary circles. For surfaces, \(\chi(M \# N) = \chi(M) + \chi(N) - 2\), and the sphere acts as the identity. See Chapter 10.

Constant loop The loop that stays at the base point. Its class is the identity of the fundamental group. See Chapter 13.

Continuous A map \(f: X \to Y\) between topological spaces is continuous if the preimage of every open set of \(Y\) is open in \(X\). For metric spaces this is the familiar \(\varepsilon\)-\(\delta\) definition. See Chapter 1 and Chapter 5.

Continuous at a point A map \(f\) is continuous at \(a\) if for every open set \(V\) containing \(f(a)\) there is an open set \(U\) containing \(a\) with \(f(U) \subseteq V\). A map is continuous when it is continuous at every point. See Chapter 5.

Contractible A space homotopy equivalent to a point: its identity map is homotopic to a constant map. Convex sets, trees and cones are contractible. See Chapter 6 and Chapter 12.

Converge (in a topological space) A sequence \(x_n\) converges to \(x\) if every open set containing \(x\) contains \(x_n\) for all large \(n\). In a space that is not Hausdorff a sequence can converge to several limits. See Chapter 4.

Convex A subset of \(\mathbb{R}^n\) that contains the segment between any two of its points. Convex sets are path connected and contractible. See Chapter 7 and Chapter 12.

Core circle The circle running down the middle of a Möbius band (or a solid torus); the band deformation retracts onto it. See Chapter 12.

Countable A set that can be listed as a sequence, finite or infinite, like \(\mathbb{N}\) or \(\mathbb{Q}\). See Chapter 4.

Covering dimension The smallest \(n\) such that every open cover of a space has a refinement in which each point lies in at most \(n + 1\) sets. For \(\mathbb{R}^n\) it is \(n\). See Chapter 11.

Covering map A continuous surjection \(p: E \to B\) such that every point of \(B\) has an evenly covered open neighbourhood. \(E\) is then a covering space of \(B\); the map \(\mathbb{R} \to S^1\), \(t \mapsto (\cos t, \sin t)\), is the basic example. See Chapter 14.

Crosscap A Möbius band sewn into a hole of a surface. A sphere with \(k\) crosscaps is the connected sum of \(k\) projective planes. See Chapter 2 and Chapter 10.

Crossing A double point of a knot diagram, marked to show which strand passes over. The crossing number of a knot is the least number of crossings in any of its diagrams. See Chapter 18.

Cut point A point of a connected space whose removal disconnects it. The number of cut points is a topological invariant, which tells the interval from the circle. See Chapter 1 and Chapter 7.

Cycle (in a graph) A closed path in a graph that does not repeat vertices: a polygon. A connected graph without cycles is a tree. See Chapter 2.

Cycle rank The number \(E - V + c\) of a graph with \(c\) components, also called its first Betti number: the number of edges that must be cut to leave a forest. A connected graph is homotopy equivalent to a wedge of that many circles. See Chapter 3 and Chapter 12.

Cycle (simplicial) An \(n\)-chain whose boundary is zero. The \(n\)-cycles form a group \(Z_n\), and every \(n\)-boundary is an \(n\)-cycle. See Chapter 16 and Chapter 16.

Cylinder The product \(S^1 \times [0, 1]\), or a square with one pair of opposite sides glued the same way. See Chapter 6 and Chapter 6.

D

Deck transformation A homeomorphism \(h: E \to E\) of a covering space with \(p \circ h = p\). For the universal cover, the deck transformations form a group isomorphic to \(\pi_1(B)\). See Chapter 14.

Deformation retraction A homotopy from the identity of \(X\) to a retraction onto a subspace \(A\), keeping \(A\) fixed (or at least inside \(A\)). \(A\) is then a deformation retract of \(X\), and the inclusion of \(A\) is a homotopy equivalence. See Chapter 12.

Degree (of a map) For \(f: S^n \to S^n\), the integer \(d\) such that \(f_*\) on \(H_n(S^n) \cong \mathbb{Z}\) is multiplication by \(d\). For a loop in the circle it is the winding number. Homotopic maps have equal degree. See Chapter 14 and Chapter 17.

Degree (of a vertex) The number of edge-ends at a vertex of a graph, a loop counting twice. See Chapter 1.

Dehn's algorithm An algorithm solving the word problem for fundamental groups of surfaces of genus at least 2: repeatedly replace a subword that is more than half of the relator by the inverse of the rest, which is shorter. See Chapter 15.

Dense A subset whose closure is the whole space, as \(\mathbb{Q}\) is in \(\mathbb{R}\). See Chapter 4.

Descartes' theorem For a polyhedral surface, the angle defects at all vertices add up to \(360^\circ \times \chi\): \(720^\circ\) for a convex polyhedron. See Chapter 2.

Determinant (of a knot) A knot invariant, equal to 3 for the trefoil and 5 for the figure-eight knot. A knot has a Fox \(p\)-colouring exactly when the prime \(p\) divides its determinant. See Chapter 18.

Diagonal The set \(\Delta = \{(x, x)\}\) in \(X \times X\). A space is Hausdorff exactly when its diagonal is closed. See Chapter 9.

Dictionary order The order on pairs given by \((a, s) < (b, t)\) if \(a < b\), or \(a = b\) and \(s < t\), as words are ordered in a dictionary. It is used to build the long line. See Chapter 9.

Diffeomorphism A smooth bijection between smooth manifolds whose inverse is also smooth. Diffeomorphic manifolds are the same as smooth manifolds. See Chapter 11 and Chapter 20.

Discharging The method of the four colour theorem's proof that moves "charge" between vertices of a plane graph to show that every such graph contains one of an unavoidable set of configurations. See Chapter 3.

Discrete metric The metric \(d(x, y) = 1\) for \(x \ne y\) and \(d(x, x) = 0\). It induces the discrete topology. See Chapter 4.

Discrete topology The topology in which every subset is open. Every function out of a discrete space is continuous. See Chapter 4.

Disjoint union The space \(X \sqcup Y\) made of a copy of \(X\) and a copy of \(Y\) side by side, a set being open when its parts in \(X\) and in \(Y\) are open. See Chapter 6.

Dual graph The graph with one vertex for each face of a plane graph and one edge across each edge. Colouring a map's countries is colouring its dual graph's vertices. See Chapter 3.

Duality (of polyhedra) The correspondence that swaps faces and vertices of a polyhedron, putting a vertex in each face. The cube and octahedron are dual, as are the dodecahedron and icosahedron; the tetrahedron is self-dual. See Chapter 2.

Dyadic rational A number of the form \(k/2^n\). Urysohn's lemma builds its function by choosing an open set for each dyadic rational in \([0, 1]\). See Chapter 9.

Dyck's theorem A handle and a crosscap together are the same as three crosscaps: \(T \# \mathbb{RP}^2 \cong \mathbb{RP}^2 \# \mathbb{RP}^2 \# \mathbb{RP}^2\). It is why non-orientable surfaces need no handles in their normal form. See Chapter 10.

E

Edge A segment joining two vertices of a graph, or a side shared by two faces of a polyhedron. See Chapter 1 and Chapter 2.

Eight model geometries Thurston's eight homogeneous 3-dimensional geometries, among them spherical, Euclidean and hyperbolic, which by geometrisation are the shapes the pieces of every 3-manifold take. See Chapter 19.

Elliptisation conjecture The statement, part of geometrisation and now proved, that a closed 3-manifold with finite fundamental group is \(S^3/\Gamma\) for a finite group \(\Gamma\) of rotations. See Chapter 19.

Embedding A continuous map that is a homeomorphism onto its image. A continuous one-to-one map from a compact space to a Hausdorff space is an embedding. See Chapter 3 and Chapter 5.

End A point of a shape with index 1: a small neighbourhood of it is a single arm, as at the tips of the letter I. See Chapter 1.

Equicontinuous A family of functions on a metric space such that for each \(\varepsilon > 0\) one \(\delta\) works for every function in the family at once. See Chapter 8.

Equivalence class The set \([x] = \{y : y \sim x\}\) of all elements related to \(x\) by an equivalence relation. The classes form the quotient set. See Chapter 6.

Equivalence relation A relation that is reflexive, symmetric and transitive. Its classes split the set into disjoint pieces. See Chapter 6.

Equivalent knots Two knots that can be deformed into each other by an ambient isotopy of space. See Chapter 15 and Chapter 18.

Euclidean metric The ordinary distance \(d_2(x, y) = \sqrt{(x_1 - y_1)^2 + \cdots + (x_n - y_n)^2}\) on \(\mathbb{R}^n\). See Chapter 4.

Euler characteristic For a surface with a cell decomposition, \(\chi = V - E + F\); for a graph, \(\chi = V - E\); for a complex, the alternating sum of numbers of cells or of Betti numbers. It is a topological invariant, \(2 - 2g\) for an orientable closed surface of genus \(g\). See Chapter 2, Chapter 3 and Chapter 16.

Euler circuit A closed walk in a graph using every edge exactly once. A connected graph has one exactly when every vertex has even degree. See Chapter 1.

Euler path A walk in a graph using every edge exactly once, also called an Euler trail. A connected graph has one exactly when it has zero or two vertices of odd degree. See Chapter 1.

Euler-Poincaré theorem The Euler characteristic of a finite complex, the alternating sum of the numbers of simplices, equals the alternating sum \(\sum (-1)^n b_n\) of its Betti numbers. So it does not depend on the triangulation. See Chapter 16.

Euler's criterion (for graphs) A connected graph has an Euler circuit exactly when every vertex has even degree, and an Euler path exactly when it has at most two vertices of odd degree. It settles the bridges of Königsberg. See Chapter 1.

Euler's formula For a connected plane graph, or a convex polyhedron, \(V - E + F = 2\). See Chapter 2 and Chapter 3.

Evenly covered An open set \(U\) of \(B\) is evenly covered by \(p: E \to B\) if \(p^{-1}(U)\) is a disjoint union of open sets, the sheets, each mapped homeomorphically onto \(U\). See Chapter 14 and Chapter 14.

Exotic \(\mathbb{R}^4\) A smooth manifold homeomorphic to \(\mathbb{R}^4\) but not diffeomorphic to it. There are uncountably many; no \(\mathbb{R}^n\) with \(n \ne 4\) has any. See Chapter 20.

Exotic sphere A smooth manifold homeomorphic to a sphere \(S^n\) but not diffeomorphic to it. Milnor found the first in dimension 7 in 1956; \(S^7\) has 28 smooth structures. See Chapter 11, Chapter 19 and Chapter 20.

Exponent sum For a word in letters \(a, b, \ldots\), the sum of the exponents of one letter. Exponent sums are unchanged by conjugation and give the abelianisation of a group. See Chapter 15.

Extreme value theorem A continuous real function on a compact space attains a maximum and a minimum. See Chapter 8.

F

Face (of a polyhedron or plane graph) A region of a polyhedron's surface bounded by edges, or a region into which a plane graph cuts the plane, including the unbounded outer face. See Chapter 2 and Chapter 3.

Face (of a simplex) A simplex spanned by some of the vertices of a given simplex. The faces of a triangle are its three edges, its three vertices and itself. See Chapter 16.

Fáry's theorem Every planar simple graph can be drawn in the plane with straight edges. See Chapter 3.

Fibre The preimage \(p^{-1}(b)\) of a point under a map, especially a covering map. For a connected covering its size is the number of sheets. See Chapter 14.

Figure eight The wedge of two circles, a space like the figure 8. Its fundamental group is the free group on two generators. See Chapter 6 and Chapter 15.

Figure-eight knot The prime knot with four crossings. It is amphichiral, and its determinant is 5. See Chapter 18.

Finer topology A topology \(\mathcal{T}'\) is finer than \(\mathcal{T}\) on the same set if \(\mathcal{T} \subseteq \mathcal{T}'\): it has more open sets. The discrete topology is the finest of all. See Chapter 4.

Finite intersection property A family of sets has it if every finite subfamily has nonempty intersection. A space is compact exactly when every family of closed sets with this property has nonempty intersection. See Chapter 8.

First countable A space in which every point has a countable family of open neighbourhoods such that every neighbourhood contains one of them. In such a space, sequences detect closure. See Chapter 4.

Five colour theorem Every planar graph can be properly coloured with five colours. Heawood proved it in 1890 from Kempe's flawed four colour argument. See Chapter 3.

Fixed point A point \(x\) with \(f(x) = x\). A space has the fixed point property if every continuous map from it to itself has one. See Chapter 7 and Chapter 12.

Four colour theorem Every planar graph, and so every map in the plane, can be properly coloured with four colours. Appel and Haken proved it with a computer in 1976. See Chapter 3.

Fox \(p\)-colouring A labelling of the arcs of a knot diagram by integers mod a prime \(p\) such that at each crossing twice the over-arc equals the sum of the two under-arcs, mod \(p\). Tricolourings are the case \(p = 3\). See Chapter 18.

Free group The group \(F(S)\) of reduced words in the letters of a set \(S\) and their inverses, multiplied by joining and reducing. A homomorphism out of it is fixed freely by saying where the generators go. It is the fundamental group of a wedge of circles. See Chapter 15.

Free product The group \(G_1 * G_2\) of reduced words alternating between elements of \(G_1\) and of \(G_2\). It is the fundamental group of a wedge of two spaces with those groups. See Chapter 15.

Freudenthal suspension theorem The homotopy groups \(\pi_{n+k}(S^n)\) stop changing as \(n\) grows, once \(n > k + 1\). The limits are the stable homotopy groups of spheres. See Chapter 20.

Functor A rule that assigns to each space an algebraic object and to each map a homomorphism, respecting identities and composition. The fundamental group and homology are functors, which is why they are topological invariants. See Chapter 13.

Fundamental group The group \(\pi_1(X, x_0)\) of path-homotopy classes of loops at \(x_0\), with concatenation as product. It is a homotopy invariant; \(\pi_1(S^1) \cong \mathbb{Z}\). See Chapter 12 and Chapter 13.

Fundamental theorem of algebra Every nonconstant complex polynomial has a root. A topological proof compares the winding numbers of large and small circles under the polynomial. See Chapter 14.

G

Galois correspondence (for coverings) Connected coverings of a nice space \(B\), up to isomorphism, correspond to subgroups of \(\pi_1(B)\) up to conjugacy; the number of sheets is the index of the subgroup. See Chapter 14.

Gauss-Bonnet theorem For a closed surface, the total Gaussian curvature equals \(2\pi\chi\). So the sign of the curvature a surface can carry everywhere is the sign of its Euler characteristic. See Chapter 2, Chapter 10 and Chapter 20.

Generalised Poincaré conjecture A closed \(n\)-manifold homotopy equivalent to \(S^n\) is homeomorphic to \(S^n\). Smale proved it for \(n \ge 5\), Freedman for \(n = 4\), and Perelman for \(n = 3\). See Chapter 19.

Generator One of the letters of a presentation \(\langle S \mid R \rangle\), or more generally one of a set of elements from which every element of a group can be built. See Chapter 15.

Genus The number of handles of an orientable closed surface: 0 for the sphere, 1 for the torus. The Euler characteristic is \(2 - 2g\). See Chapter 2, Chapter 10 and Chapter 10.

Geometrisation conjecture Thurston's statement that every closed orientable 3-manifold can be cut along spheres and incompressible tori into pieces each carrying one of eight model geometries. Perelman proved it, and the Poincaré conjecture with it. See Chapter 11 and Chapter 19.

Girth The length of the shortest cycle in a graph. A planar graph of girth \(k\) has few edges, which is how \(K_{3,3}\) is shown not planar. See Chapter 3.

Graph A set of vertices with a set of edges, each joining two vertices. Seen as a space, a graph is a union of points and arcs. See Chapter 1 and Chapter 3.

Graph map The map \(x \mapsto (x, g(x))\) from \(X\) to \(X \times Y\) for a continuous \(g: X \to Y\). It is an embedding of \(X\) onto the graph of \(g\). See Chapter 6.

H

Hairy ball theorem An even-dimensional sphere has no continuous nowhere-zero tangent vector field: you cannot comb a hairy ball flat. On Earth, the wind is calm somewhere. See Chapter 17.

Haken manifold A compact irreducible 3-manifold containing an incompressible surface. Thurston proved geometrisation for these before the general case. See Chapter 19.

Ham sandwich theorem Any \(n\) bounded measurable sets in \(\mathbb{R}^n\) can be cut in half simultaneously by one hyperplane. It follows from the Borsuk-Ulam theorem. See Chapter 17.

Handle A thickened disc \(D^k \times D^{n-k}\) attached to a manifold along \(\partial D^k \times D^{n-k}\). Every smooth closed manifold is built from a ball by attaching handles. See Chapter 19.

Hausdorff dimension A dimension defined from how many small sets are needed to cover a space, which can be a fraction. It is not a topological invariant: the Koch snowflake has Hausdorff dimension about 1.26. See Chapter 11.

Hausdorff space A space in which any two different points have disjoint open neighbourhoods, also called T2. In a Hausdorff space limits of sequences are unique. See Chapter 4 and Chapter 9.

h-cobordism theorem Smale's theorem that a simply connected h-cobordism between manifolds of dimension at least 5 is a product. It proves the Poincaré conjecture in dimensions 5 and up. See Chapter 19.

Heawood's bound The number of colours needed for maps on a closed surface of Euler characteristic \(\chi \le 0\) is at most \(\lfloor (7 + \sqrt{49 - 24\chi})/2 \rfloor\): seven for the torus. See Chapter 3.

Heine-Borel theorem A subset of \(\mathbb{R}^n\) is compact exactly when it is closed and bounded. See Chapter 8.

Heine-Cantor theorem A continuous map from a compact metric space to a metric space is uniformly continuous. See Chapter 8.

Higher homotopy group The group \(\pi_n(X)\) of homotopy classes of maps \(S^n \to X\) sending a base point to a base point. For \(n \ge 2\) it is abelian. See Chapter 16 and Chapter 20.

Hilbert cube The product \([0, 1]^{\mathbb{N}}\) of countably many intervals, compact by Tychonoff. Every compact metric space embeds in it. See Chapter 8 and Chapter 9.

Hilbert-Smith conjecture The conjecture that no \(p\)-adic group acts faithfully on a manifold. Pardon proved it in dimension 3 in 2013. See Chapter 20.

Homeomorphic Two spaces are homeomorphic if there is a homeomorphism between them, written \(X \cong Y\). This is topology's "the same". See Chapter 1 and Chapter 5.

Homeomorphism A continuous bijection whose inverse is also continuous: a deformation of one space onto another with no tearing or gluing. See Chapter 1 and Chapter 5.

Homogeneous coordinates The notation \([x_0 : \cdots : x_n]\) for the point of \(\mathbb{RP}^n\) given by the line through \((x_0, \ldots, x_n)\); scaling all coordinates gives the same point. See Chapter 11.

Homologous Two \(n\)-cycles are homologous if their difference is a boundary. They then give the same homology class. See Chapter 16.

Homology group The group \(H_n = Z_n / B_n\) of \(n\)-cycles modulo \(n\)-boundaries. It counts \(n\)-dimensional holes: \(H_0\) counts components, \(H_1\) is the abelianised fundamental group of a path-connected space. See Chapter 5 and Chapter 16.

Homotopic Two maps \(f, g: X \to Y\) are homotopic if one can be continuously deformed into the other through a homotopy. See Chapter 12.

Homotopy A continuous map \(H: X \times [0, 1] \to Y\), thought of as a family of maps \(H(\cdot, t)\) moving from \(f = H(\cdot, 0)\) to \(g = H(\cdot, 1)\). See Chapter 12.

Homotopy class The set of all maps homotopic to a given map, or of all paths path-homotopic to a given path. See Chapter 12 and Chapter 13.

Homotopy equivalence A map \(f: X \to Y\) with a homotopy inverse \(g: Y \to X\), meaning \(g \circ f\) and \(f \circ g\) are homotopic to the identities. Spaces related this way are homotopy equivalent, or have the same homotopy type. See Chapter 12.

Homotopy invariant A property or quantity that is the same for homotopy equivalent spaces, such as the fundamental group, homology and the Euler characteristic. See Chapter 12.

Homotopy lifting For a covering map, a homotopy of paths in the base lifts uniquely to a homotopy upstairs once the starting lift is chosen. It is how the covering of the circle computes \(\pi_1(S^1)\). See Chapter 14 and Chapter 14.

Homotopy type The class of all spaces homotopy equivalent to a given space. The letters of the alphabet have three homotopy types: a point, a circle and a figure eight. See Chapter 12.

Hopf fibration The map \(H: S^3 \to S^2\) whose fibres are circles, any two of them linked once. It is not null-homotopic, so \(\pi_3(S^2)\) is not trivial. See Chapter 20.

Hopf invariant For a smooth map \(f: S^3 \to S^2\), the linking number of the preimages of two different regular values. The Hopf map has Hopf invariant 1. See Chapter 20.

Hopf invariant one problem Which spheres \(S^n\) admit maps \(S^{2n-1} \to S^n\) of Hopf invariant one? Adams showed in 1960 that only \(n = 2, 4, 8\) do. See Chapter 20.

Hopf trace formula For a map of a finite complex, the alternating sum of the traces on the chain groups equals the alternating sum of the traces on homology. It proves the Lefschetz fixed point theorem. See Chapter 17.

Hopf's theorem (on degree) Two maps \(S^n \to S^n\) are homotopic exactly when they have the same degree. See Chapter 17.

Hurewicz theorem For a path-connected space, \(H_1\) is the abelianisation of \(\pi_1\). The higher version links the first nonzero homotopy and homology groups of a simply connected space. See Chapter 16.

Hyperbolic Having constant curvature \(-1\). Closed surfaces with \(\chi < 0\) are hyperbolic, and by geometrisation most 3-manifolds are. See Chapter 19.

I

Incompressible surface A surface in a 3-manifold whose fundamental group injects into that of the manifold, so its loops cannot be squeezed to points through the manifold. Geometrisation cuts along incompressible tori. See Chapter 19.

Independent cycles The cycles of a graph that close up as each edge outside a spanning tree is added back. Their number is the cycle rank. See Chapter 3.

Index (of a point) A point of a shape has index \(k\) if a small neighbourhood of it looks like a star with \(k\) arms: ends have index 1, points inside a stroke index 2, branch points 3 or more. See Chapter 1.

Index (of a vector field zero) The degree of the direction of a vector field on a small circle around an isolated zero. The Poincaré-Hopf theorem says the indices add up to the Euler characteristic. See Chapter 17.

Indiscrete topology The topology on a set whose only open sets are the empty set and the whole set, also called the trivial topology. See Chapter 4.

Induced homomorphism The homomorphism \(f_*: \pi_1(X, x_0) \to \pi_1(Y, f(x_0))\) given by \(f_*[\gamma] = [f \circ \gamma]\), and its analogue on homology. It satisfies \((g \circ f)_* = g_* \circ f_*\) and \(\mathrm{id}_* = \mathrm{id}\). See Chapter 13.

Inductive dimension A dimension defined by induction: a space has dimension at most \(n\) if every point has small neighbourhoods whose boundaries have dimension at most \(n - 1\). See Chapter 11.

Interior The largest open set inside a set \(A\), written \(\operatorname{int} A\): the union of all open sets contained in \(A\). See Chapter 4.

Interior (of a manifold with boundary) The points of a manifold with boundary that have neighbourhoods homeomorphic to \(\mathbb{R}^n\). See Chapter 10.

Intermediate value theorem A continuous real function on an interval takes every value between any two of its values. It holds because intervals are connected. See Chapter 7.

Intrinsic and extrinsic Intrinsic questions ask about a space alone (is \(X\) homeomorphic to \(Y\)?); extrinsic questions ask how it sits inside a larger space (can one embedding be moved to another?). See Chapter 1.

Invariance of dimension If \(m \ne n\), no nonempty open subset of \(\mathbb{R}^m\) is homeomorphic to an open subset of \(\mathbb{R}^n\). Brouwer proved it from invariance of domain; it makes the dimension of a manifold well defined. See Chapter 11.

Invariance of domain Brouwer's theorem that a continuous one-to-one map from an open set of \(\mathbb{R}^n\) to \(\mathbb{R}^n\) has open image. It implies that \(\mathbb{R}^m\) and \(\mathbb{R}^n\) are not homeomorphic for \(m \ne n\). See Chapter 11.

Isolated point A point \(x\) with \(\{x\}\) open. Their number is a topological invariant. See Chapter 5.

Isomorphism problem The problem of deciding, from two finite presentations, whether the groups are isomorphic. Like the word problem it is unsolvable in general. See Chapter 15.

Isotopy A continuous family of embeddings, moving one embedding of a space to another without passing through itself. See Chapter 1.

J

Jones polynomial A Laurent polynomial invariant of oriented knots and links found by Vaughan Jones in 1984, computed from the Kauffman bracket and the writhe. It tells the trefoil from its mirror image. See Chapter 18.

Jordan curve theorem A simple closed curve in the plane separates it into exactly two components, a bounded inside and an unbounded outside, and is the boundary of each. See Chapter 1.

Jordan-Schoenflies theorem The inside of a simple closed curve in the plane, together with the curve, is homeomorphic to a closed disc. See Chapter 1.

K

Kempe chain In a coloured graph, a maximal connected set of vertices using only two given colours. Swapping the two colours on it keeps the colouring proper. See Chapter 3.

Kervaire invariant problem The question in which dimensions certain elements of the stable homotopy groups of spheres exist. Hill, Hopkins and Ravenel settled it in all dimensions except 126. See Chapter 20.

Khovanov homology A homology theory of knots whose Euler characteristic is the Jones polynomial. It detects the unknot. See Chapter 18.

Klein bottle The closed non-orientable surface made from a square by gluing one pair of opposite sides the same way and the other pair with a flip, word \(abab^{-1}\). It cannot be embedded in \(\mathbb{R}^3\), and \(\chi = 0\). See Chapter 2 and Chapter 6.

Knot A subset of \(\mathbb{R}^3\) (or \(S^3\)) homeomorphic to a circle, considered up to ambient isotopy. See Chapter 15 and Chapter 18.

Knot diagram A regular projection of a knot to a plane, with each crossing marked to show which strand is on top. See Chapter 18.

Knot group The fundamental group of the complement of a knot in space. Equivalent knots have isomorphic knot groups; the unknot's is \(\mathbb{Z}\). See Chapter 15 and Chapter 18.

Koch snowflake A fractal simple closed curve made by repeatedly adding triangular bumps to the sides of a triangle. It has infinite length yet bounds a finite area. See Chapter 1.

K-topology The topology on \(\mathbb{R}\) with basis the open intervals and the sets \((a, b) \setminus K\), where \(K = \{1/n\}\). It is Hausdorff but not regular. See Chapter 9.

Kuratowski's theorem A graph is planar exactly when it contains no subdivision of \(K_5\) or \(K_{3,3}\). See Chapter 3.

L

Lebesgue number For an open cover of a compact metric space, a number \(\delta > 0\) such that every set of diameter less than \(\delta\) lies inside one member of the cover. See Chapter 8 and Chapter 14.

Lefschetz fixed point theorem For a map \(f\) of a finite complex (or a compact triangulable space), if the Lefschetz number \(L(f) \ne 0\) then \(f\) has a fixed point. See Chapter 17.

Lefschetz number The number \(L(f) = \sum_k (-1)^k \operatorname{tr}(f_* \text{ on } H_k(X; \mathbb{Q}))\). For the identity it is the Euler characteristic. See Chapter 17.

Lens space The 3-manifold \(L(p, q)\), the quotient of \(S^3\) by a rotation of order \(p\). It has fundamental group \(\mathbb{Z}/p\); \(L(2, 1) = \mathbb{RP}^3\). See Chapter 11 and Chapter 19.

Lift For a covering \(p: E \to B\), a lift of \(f: Y \to B\) is a map \(\tilde f: Y \to E\) with \(p \circ \tilde f = f\). See Chapter 14.

Lifting criterion A map \(f: Y \to B\) from a nice connected space lifts to a covering \(E\) exactly when \(f_*\pi_1(Y)\) lies in \(p_*\pi_1(E)\). In particular every map from a simply connected space lifts. See Chapter 14.

Limit point A point \(x\) is a limit point of \(A\) if every open set around \(x\) meets \(A\) in a point other than \(x\). The closure of \(A\) is \(A\) together with its limit points. See Chapter 4.

Link A finite collection of disjoint knots, its components. See Chapter 18.

Linking number For a two-component oriented link, half the sum of the signs of the crossings between the two components. It is an invariant: the Hopf link has linking number \(\pm 1\). See Chapter 18 and Chapter 18.

Locally compact A space in which every point has an open neighbourhood contained in a compact set. A Hausdorff space has a Hausdorff one-point compactification exactly when it is locally compact. See Chapter 8.

Locally path connected A space in which every neighbourhood of a point contains a path-connected open neighbourhood. In such a space components and path components coincide. See Chapter 7.

Long line Two copies of the long ray glued at their ends: a connected Hausdorff space that is locally like \(\mathbb{R}\) yet not a manifold, since it is not second countable. See Chapter 9.

Long ray The set \(\omega_1 \times [0, 1)\) in the dictionary order topology (the closed long ray), or that set without its first point (the open long ray): uncountably many intervals laid end to end. See Chapter 9.

Loop A path that starts and ends at the same point, the base point. In a graph, a loop is an edge from a vertex to itself. See Chapter 3 and Chapter 13.

Lower limit topology The topology on \(\mathbb{R}\) with basis the half-open intervals \([a, b)\). With it \(\mathbb{R}\) is the Sorgenfrey line. See Chapter 4.

Lusternik-Schnirelmann theorem If the sphere is covered by three closed sets, one of them contains a pair of antipodal points. It follows from the Borsuk-Ulam theorem in dimension 2. See Chapter 14.

M

Manifold A topological \(n\)-manifold is a Hausdorff, second countable space in which every point has a neighbourhood homeomorphic to \(\mathbb{R}^n\). Curves are 1-manifolds and surfaces 2-manifolds. See Chapter 11.

Manifold with boundary A space locally like the half-space \(\{x_n \ge 0\}\) in \(\mathbb{R}^n\), such as a disc or a solid ball. Its boundary is the set of points on the edge. See Chapter 10 and Chapter 11.

Mapping cylinder The space \(M_f\) made by gluing the end \(X \times \{1\}\) of a cylinder \(X \times [0, 1]\) to \(Y\) by \(f\). It deformation retracts onto \(Y\). See Chapter 12.

Maximum metric The metric \(d_\infty(x, y) = \max_i |x_i - y_i|\) on \(\mathbb{R}^n\). It gives the same open sets as the Euclidean metric. See Chapter 4.

Mayer-Vietoris sequence A long exact sequence that computes the homology of \(A \cup B\) from those of \(A\), \(B\) and \(A \cap B\): the homology analogue of the Seifert-van Kampen theorem. See Chapter 16.

Meridian For a knot, a small loop going once around the knot, generating the knot group up to conjugacy. For a curve in the plane, a test line used to count crossings. See Chapter 1 and Chapter 15.

Metric A distance function \(d\) on a set with \(d(x, y) \ge 0\), equal to 0 exactly when \(x = y\), symmetric, and satisfying the triangle inequality. See Chapter 4.

Metric space A set with a metric. Its open sets, unions of open balls, form the metric topology. See Chapter 4.

Metric topology The topology on a metric space whose open sets are the unions of open balls. See Chapter 4.

Metrisable A topological space whose topology comes from some metric. Metrisable spaces are Hausdorff and normal. See Chapter 4 and Chapter 9.

Minor A graph obtained from a subgraph by contracting edges. Wagner's theorem: a graph is planar exactly when neither \(K_5\) nor \(K_{3,3}\) is a minor. See Chapter 3.

Möbius strip The surface made from a strip by gluing its ends with a half twist. It has one side and one edge, and is the simplest non-orientable surface. Also called the Möbius band. See Chapter 2, Chapter 6 and Chapter 10.

Moment curve The curve \(t \mapsto (t, t^2, t^3)\) in \(\mathbb{R}^3\). Putting vertices on it shows every graph embeds in \(\mathbb{R}^3\) with straight edges. See Chapter 3.

Multiple edges Two or more edges joining the same pair of vertices of a graph. A simple graph has none, and no loops. See Chapter 3.

N

Nagata-Smirnov metrisation theorem A space is metrisable exactly when it is regular and has a countably locally finite basis. See Chapter 9.

Neck pinch A singularity of Ricci flow where a thin neck between two bulbs of a 3-manifold shrinks to a point. Perelman's surgery cuts there and caps the ends. See Chapter 19.

Necklace splitting theorem A necklace with beads of \(k\) colours, each appearing an even number of times, can be shared fairly between two thieves using at most \(k\) cuts. Alon proved it with the Borsuk-Ulam theorem. See Chapter 17.

Nielsen-Schreier theorem Every subgroup of a free group is free. It follows because covering spaces of graphs are graphs. See Chapter 15.

Non-orientable A surface containing a Möbius band, so that some loop reverses clockwise. The Klein bottle and the projective plane are non-orientable. See Chapter 2 and Chapter 10.

Normal closure The smallest normal subgroup of a group containing a given set of elements. A presentation's group is the free group modulo the normal closure of its relators. See Chapter 15.

Normal form (of a surface word) One of the standard words \(aa^{-1}\) (sphere), \(a_1b_1a_1^{-1}b_1^{-1} \cdots a_gb_ga_g^{-1}b_g^{-1}\) (genus \(g\)) or \(a_1a_1 \cdots a_ka_k\) (\(k\) crosscaps), to which every labelled polygon can be reduced. See Chapter 10.

Normal space A T1 space in which any two disjoint closed sets have disjoint open neighbourhoods, also called T4. Metric spaces and compact Hausdorff spaces are normal. See Chapter 9.

Normal surface A surface in a triangulated 3-manifold meeting each tetrahedron in triangles and quadrilaterals. Haken's normal surfaces give an algorithm to recognise the unknot. See Chapter 18.

NP and co-NP The classes of decision problems whose yes-answers, or no-answers, have certificates checkable in polynomial time. Unknot recognition lies in both. See Chapter 18.

Null-homotopic A map homotopic to a constant map. See Chapter 12.

O

Odd map A map of spheres with \(f(-x) = -f(x)\). An odd map \(S^n \to S^n\) has odd degree, which proves the Borsuk-Ulam theorem. See Chapter 14 and Chapter 17.

One-point compactification The space \(X^\ast = X \cup \{\infty\}\) whose open sets are those of \(X\) and the complements of compact closed subsets of \(X\) with \(\infty\) added. The one-point compactification of \(\mathbb{R}^n\) is \(S^n\). See Chapter 8 and Chapter 11.

Open ball In a metric space, the set \(B(x, r) = \{y : d(x, y) < r\}\) of points closer than \(r\) to \(x\). See Chapter 4.

Open cover A collection of open sets whose union is the whole space (or contains a given set). A subcollection that still covers is a subcover. See Chapter 8.

Open map A map that sends open sets to open sets. Projections from products are open, and homeomorphisms are exactly the open continuous bijections. See Chapter 5 and Chapter 6.

Open neighbourhood An open set containing a given point. See Chapter 4.

Open set In a metric space, a set containing an open ball around each of its points; in general, a member of the chosen topology. See Chapter 1, Chapter 4 and Chapter 4.

Order topology The topology on an ordered set generated by the open rays \(\{x < a\}\) and \(\{x > a\}\). On \(\mathbb{R}\) it is the usual topology. See Chapter 4.

Orientable A surface in which no closed path reverses clockwise; equivalently, one that contains no Möbius band. The sphere and tori are orientable. See Chapter 2 and Chapter 10.

Orientation (of a knot) A chosen direction of travel along a knot. With it each crossing gets a sign, used for the writhe and the linking number. See Chapter 18.

Orientation (of a simplex) A choice of ordering of the vertices of a simplex up to even permutations. Swapping two vertices reverses it, which is what the signs in the boundary map record. See Chapter 16.

Orientation-reversing loop A loop on a surface along which a small circle with a direction comes back with the opposite direction, as around the core of a Möbius strip. See Chapter 2.

Outer face The unbounded face of a plane graph. See Chapter 3.

P

Parity Whether a number is even or odd. A point is inside a polygon when a ray from it crosses the polygon an odd number of times. See Chapter 1.

Partition of unity A family of continuous functions to \([0, 1]\) that add up to 1, each vanishing outside a chosen open set. Urysohn's lemma provides them on normal spaces. See Chapter 9.

Pasting lemma If \(X = A \cup B\) with \(A\) and \(B\) closed (or both open), and continuous maps on \(A\) and on \(B\) agree on \(A \cap B\), they glue to a continuous map on \(X\). See Chapter 5, Chapter 12 and Chapter 13.

Path A continuous map \(\gamma: [0, 1] \to X\), starting at \(\gamma(0)\) and ending at \(\gamma(1)\). See Chapter 6, Chapter 7 and Chapter 13.

Path class The set of all paths path homotopic to a given path, written \([\gamma]\). Path classes of loops are the elements of the fundamental group. See Chapter 13.

Path component A maximal path-connected subset of a space. The path components of \(X\) correspond to the elements of \(H_0(X)\). See Chapter 7.

Path connected A space in which any two points are joined by a path. Path-connected spaces are connected; the topologist's sine curve is connected but not path connected. See Chapter 7.

Path homotopy A homotopy between two paths that keeps both endpoints fixed throughout. Paths related this way are path homotopic. See Chapter 12 and Chapter 13.

Path lifting For a covering map, every path in the base lifts uniquely to a path upstairs starting at any chosen point over its start. See Chapter 14 and Chapter 14.

PD code A way to store a knot diagram as a list: number the edges \(1, \ldots, 2n\) along the knot and record each crossing as its four edge numbers. See Chapter 18.

Perfect group A group equal to its commutator subgroup, so its abelianisation is trivial. The binary icosahedral group is perfect, which is why the Poincaré homology sphere has the homology of \(S^3\). See Chapter 19.

Persistent homology A method of topological data analysis that computes homology of a point cloud at every scale at once and records the births and deaths of features as a barcode. See Chapter 17.

Petersen graph A 3-regular graph with 10 vertices and 15 edges and girth 5. It is not planar. See Chapter 3.

Plain pair and twisted pair In a word for a labelled polygon, a letter appearing as \(a \ldots a^{-1}\) is a plain pair, and as \(a \ldots a\) a twisted pair. A twisted pair makes a Möbius band, so the surface is non-orientable. See Chapter 10.

Planar graph A graph that can be drawn in the plane with no edges crossing. Such a drawing is a plane graph. See Chapter 3.

Plane graph A graph drawn in the plane with no edges crossing. It cuts the plane into faces. See Chapter 2 and Chapter 3.

Platonic solid A convex regular polyhedron: the tetrahedron, cube, octahedron, dodecahedron or icosahedron. Euler's formula shows there are no others. See Chapter 2.

Poincaré conjecture Every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. Poincaré asked it in 1904 and Perelman proved it in 2002 to 2003 with Ricci flow. See Chapter 11 and Chapter 19.

Poincaré homology sphere A closed 3-manifold with the same homology as \(S^3\) but fundamental group the binary icosahedral group of order 120. It made Poincaré add the fundamental group to his conjecture. See Chapter 11, Chapter 16 and Chapter 19.

Poincaré-Hopf theorem For a vector field with isolated zeros on a closed manifold, the sum of the indices of the zeros equals the Euler characteristic. See Chapter 17.

Point cloud A finite set of points in a metric space, the raw input of topological data analysis. See Chapter 17.

Polygon A closed path of straight segments in the plane that does not cross itself. See Chapter 1.

Polyhedron A solid bounded by flat polygonal faces, meeting in edges and vertices. In Chapter 16, the space \(|K|\) of a simplicial complex. See Chapter 2 and Chapter 16.

Preimage For \(f: X \to Y\) and \(V \subseteq Y\), the set \(f^{-1}(V) = \{x : f(x) \in V\}\). Preimages respect unions, intersections and complements, which is why continuity is defined with them. See Chapter 5.

Presentation A description \(\langle S \mid R \rangle\) of a group by generators \(S\) and relators \(R\): the free group on \(S\) modulo the normal closure of \(R\). See Chapter 15.

Prime 3-manifold A 3-manifold that is not a connected sum except trivially with \(S^3\). Every closed orientable 3-manifold is a connected sum of primes. See Chapter 19.

Prime knot A non-trivial knot that is not a connected sum of two non-trivial knots. Every knot is a connected sum of prime knots in essentially one way. See Chapter 18.

Product topology The topology on a product \(\prod X_i\) generated by the sets \(\pi_i^{-1}(U)\) for \(U\) open in \(X_i\). For two factors its basis is the open rectangles \(U \times V\); it is the coarsest topology making all projections continuous. See Chapter 6.

Projection The map \(\pi_i: \prod X_j \to X_i\) that keeps the \(i\)-th coordinate. Projections are continuous and open. See Chapter 6.

Projective plane The closed non-orientable surface \(\mathbb{RP}^2\): the space of lines through the origin in \(\mathbb{R}^3\), a sphere with antipodal points glued, or a disc with opposite rim points glued (word \(abab\)). \(\chi = 1\) and \(\pi_1 = \mathbb{Z}/2\). See Chapter 2, Chapter 6 and Chapter 15.

Q

Quotient map A surjective map \(q: X \to Y\) such that a set \(V \subseteq Y\) is open exactly when \(q^{-1}(V)\) is open. See Chapter 6.

Quotient topology The finest topology on a quotient set making the quotient map continuous: a set of classes is open when the union of its classes is open. It makes gluing precise. See Chapter 6.

R

Radó's theorem Every surface can be triangulated. It is the foundation of the combinatorial classification of surfaces. See Chapter 10.

Rank (of a free group) The number of generators of a free group \(F_n\). Free groups of different ranks are not isomorphic. See Chapter 15.

Real projective space The space \(\mathbb{RP}^n\) of lines through the origin in \(\mathbb{R}^{n+1}\), equivalently \(S^n\) with antipodal points glued. It is an \(n\)-manifold. See Chapter 6 and Chapter 11.

Reduced word A word in which no letter stands next to its own inverse. Every word reduces to a unique reduced word. See Chapter 15.

Reducible configuration A configuration of a plane graph that cannot occur in a smallest counterexample to the four colour theorem, because any colouring of the rest extends. See Chapter 3.

Refinement (of a decomposition) A cell decomposition obtained from another by subdividing its cells. Any two decompositions of a surface have a common refinement, which is why the Euler characteristic is well defined. See Chapter 2.

Regular polyhedron A polyhedron whose faces are all the same regular polygon, with the same number meeting at each vertex. See Chapter 2.

Regular projection A projection of a knot with only finitely many double points, no triple points, and transverse crossings. See Chapter 18.

Regular space A T1 space in which a closed set and a point outside it have disjoint open neighbourhoods, also called T3. See Chapter 9.

Regular value A point \(y\) such that the derivative of \(f\) is onto at every point of \(f^{-1}(y)\). By the regular value theorem the preimage is then a smooth manifold, and the degree can be counted at its preimage points. See Chapter 11 and Chapter 17.

Reidemeister moves The three local moves on knot diagrams: R1 (twist), R2 (poke) and R3 (slide). Reidemeister's theorem: two diagrams show equivalent knots exactly when a sequence of these moves joins them. See Chapter 18.

Relation (in a presentation) An equation that the generators of a presented group must satisfy. A relator is a word required to equal 1. See Chapter 15.

Relative homotopy A homotopy relative to \(A\) keeps every point of \(A\) fixed throughout. See Chapter 12.

Relatively open A subset of \(Y \subseteq X\) that is open in the subspace topology of \(Y\), though possibly not open in \(X\). See Chapter 4.

Retraction A continuous map \(r: X \to A\) onto a subspace with \(r(a) = a\) for every \(a \in A\). \(A\) is then a retract of \(X\). There is no retraction of a disc onto its boundary circle. See Chapter 12, Chapter 13 and Chapter 17.

Ricci curvature A measure of how a Riemannian metric makes volumes of small balls differ from flat space, in each direction. See Chapter 19.

Ricci flow Hamilton's equation that changes a Riemannian metric by minus twice its Ricci curvature, like a heat equation for shape. Perelman used it, with surgery, to prove geometrisation. See Chapter 19.

Riemann sphere The complex plane with one point at infinity added, the one-point compactification of \(\mathbb{C}\), homeomorphic to \(S^2\). See Chapter 8.

Riemannian metric A smoothly varying inner product on the tangent spaces of a smooth manifold, which gives lengths, angles and curvature. See Chapter 19.

Rotation group The group \(SO(3)\) of rotations of \(\mathbb{R}^3\), a 3-manifold homeomorphic to \(\mathbb{RP}^3\). See Chapter 11.

Rubber-sheet geometry A popular name for topology: the study of properties that survive stretching and bending but not tearing or gluing. See Chapter 1.

S

Schoenflies problem (4D) The open question whether every smooth 3-sphere in \(S^4\) bounds smooth 4-balls on both sides. See Chapter 20 and Chapter 20.

Second countable A space with a countable basis. \(\mathbb{R}^n\) is second countable; manifolds are required to be. See Chapter 4, Chapter 5 and Chapter 9.

Seifert surface A compact orientable surface in space whose boundary is a given knot. Seifert's theorem says every knot has one. See Chapter 10.

Seifert-van Kampen theorem If \(X = U \cup V\) with \(U\), \(V\) and \(U \cap V\) open and path connected, then \(\pi_1(X)\) is the amalgamated free product of \(\pi_1(U)\) and \(\pi_1(V)\) over \(\pi_1(U \cap V)\). See Chapter 15.

Separable A space with a countable dense subset, as \(\mathbb{R}\) has \(\mathbb{Q}\). See Chapter 4 and Chapter 5.

Separation A pair of disjoint nonempty open sets whose union is the whole space. A space is connected when it has none. See Chapter 7.

Sequentially compact A space in which every sequence has a convergent subsequence. For metric spaces it is equivalent to compactness. See Chapter 8 and Chapter 9.

Sheet One of the disjoint open sets over an evenly covered set \(U\), each mapped homeomorphically onto \(U\) by the covering map. See Chapter 14 and Chapter 14.

Sierpiński space The two-point space \(\{0, 1\}\) whose open sets are \(\varnothing\), \(\{1\}\) and \(\{0, 1\}\). It is T0 but not T1. See Chapter 4 and Chapter 9.

Simple closed curve A subset of the plane homeomorphic to a circle: a closed curve that does not cross itself. See Chapter 1.

Simple graph A graph with no loops and no multiple edges. See Chapter 3.

Simplex The \(n\)-simplex is the convex hull of \(n + 1\) affinely independent points: a point, a segment, a triangle, a tetrahedron, and so on. See Chapter 16.

Simplicial approximation theorem Every continuous map between polyhedra is homotopic to a simplicial map after subdividing the domain enough. It makes homology a topological invariant. See Chapter 17.

Simplicial complex A finite collection of simplices in some \(\mathbb{R}^N\) closed under taking faces, any two meeting in a common face or not at all. See Chapter 16.

Simply connected A path-connected space with trivial fundamental group: every loop shrinks to a point. Convex sets and \(S^n\) for \(n \ge 2\) are simply connected. See Chapter 13 and Chapter 19.

Singular homology A homology defined for every space using all continuous maps of simplices into it, agreeing with simplicial homology on simplicial complexes. See Chapter 16.

Skein relation A linear relation among the Jones polynomials of three links that differ at one crossing. It computes the polynomial by induction. See Chapter 18.

Slice knot A knot in the 3-sphere that bounds a smooth disc in the 4-ball. Piccirillo proved in 2020 that the Conway knot is not slice. See Chapter 20.

Slice-ribbon conjecture The open conjecture that every slice knot is ribbon, bounding a disc of a special kind with only simple self-intersections. See Chapter 20.

Smith normal form A diagonal form of an integer matrix reached by integer row and column operations. Its diagonal gives the torsion of a homology group or of an abelianised presentation. See Chapter 15 and Chapter 16.

Smooth 4-dimensional Poincaré conjecture The open conjecture that every smooth 4-manifold homeomorphic to \(S^4\) is diffeomorphic to it. See Chapter 20 and Chapter 20.

Smooth atlas An atlas whose transition maps are all smooth, so that its charts are smoothly compatible. See Chapter 11.

Smooth manifold A manifold with a maximal smooth atlas. Maps between smooth manifolds are smooth when they are smooth in charts. See Chapter 11.

Smooth structure A choice of maximal smooth atlas on a topological manifold. A manifold can have none, one, or many. See Chapter 20.

Solid torus The product \(D^2 \times S^1\), a doughnut including its inside. See Chapter 6 and Chapter 11.

Sorgenfrey line The real line with the lower limit topology. It is normal and first countable but not second countable, and the product of two copies, the Sorgenfrey plane, is not normal. See Chapter 4, Chapter 9 and Chapter 9.

Space-filling curve A continuous map from \([0, 1]\) onto the square, such as Peano's or Hilbert's. It shows a continuous map can raise dimension, and that a loop need not miss any point. See Chapter 11 and Chapter 13.

Spanning tree A tree in a graph containing all its vertices. Contracting it shows a connected graph is homotopy equivalent to a wedge of circles. See Chapter 3, Chapter 12 and Chapter 15.

Sperner's lemma In a suitably labelled triangulation of a triangle, some small triangle carries all three labels. It gives a constructive route to Brouwer's fixed point theorem. See Chapter 14.

Sphere The \(n\)-sphere \(S^n\) is the set of points at distance 1 from the origin in \(\mathbb{R}^{n+1}\). \(S^1\) is the circle and \(S^2\) the ordinary sphere; \(S^3\) is the 3-sphere, the simplest closed 3-manifold. See Chapter 6, Chapter 11 and Chapter 19.

Stability theorem (persistent homology) Moving each point of a cloud by at most \(\varepsilon\) moves the barcode by at most \(\varepsilon\) in the bottleneck distance, so persistent homology is robust to noise. See Chapter 17.

Stable homotopy groups of spheres The limits \(\pi_k^s\) of \(\pi_{n+k}(S^n)\) as \(n\) grows. No general formula is known. See Chapter 20.

Standard simplex The simplex \(\{x \in \mathbb{R}^n : x_i \ge 0, \sum x_i = 1\}\), whose points are probability vectors. See Chapter 17.

Star-shaped A set with a point from which the segment to every other point stays in the set. Star-shaped sets are contractible and simply connected. See Chapter 12 and Chapter 13.

State sum The expansion of the Kauffman bracket as a sum over states, a state being a choice of A- or B-smoothing at every crossing. See Chapter 18.

Stereographic projection The map from a sphere minus its north pole to the plane, sending each point to where the line from the north pole through it meets the plane. It is a homeomorphism. See Chapter 1, Chapter 5 and Chapter 11.

Subbasis A collection of open sets whose finite intersections form a basis. The open rays form a subbasis for the order topology. See Chapter 4.

Subdivision (of a graph) A graph obtained by putting new vertices of degree 2 in the middle of edges. Subdivisions are homeomorphic as spaces. See Chapter 3 and Chapter 3.

Subspace topology The topology on a subset \(Y\) of \(X\) whose open sets are the intersections \(U \cap Y\) with \(U\) open in \(X\). See Chapter 4.

Surface A space that looks like the plane near each point: a Hausdorff, second countable space in which every point has a neighbourhood homeomorphic to an open disc. See Chapter 2 and Chapter 10.

Surface with boundary A space in which each point has a neighbourhood homeomorphic to an open disc or to a half-disc. Every compact one is a closed surface with some open discs removed. See Chapter 10.

Surgery Cutting a manifold along a sphere and gluing in balls. Perelman used surgery to continue Ricci flow past neck pinches. See Chapter 19.

Suspension The space \(SX\) made from \(X \times [0, 1]\) by collapsing \(X \times \{0\}\) to one point and \(X \times \{1\}\) to another: a double cone. The suspension of \(S^n\) is \(S^{n+1}\). See Chapter 6.

T

T0, T1, T2, T3, T4 The separation axioms. T0: the topology tells points apart. T1: points are closed. T2: Hausdorff. T3: regular. T4: normal. See Chapter 9 and Chapter 9.

Tame knot A knot equivalent to a closed polygon with finitely many sides. Smooth knots are tame; knots that are not are wild. See Chapter 18.

Tangent vector field A continuous choice of a tangent vector at each point of a manifold. See Chapter 17.

Taxicab metric The metric \(d_1(x, y) = |x_1 - y_1| + \cdots + |x_n - y_n|\) on \(\mathbb{R}^n\), distance along a grid of streets. See Chapter 4.

Theta graph The graph with two vertices joined by three edges. Its fundamental group is the free group of rank 2. See Chapter 15.

Tietze extension theorem On a normal space, every continuous real function on a closed subset extends to a continuous function on the whole space. See Chapter 9.

Topoisomerase An enzyme that changes the topology of DNA by cutting strands and passing them through each other, changing linking number. See Chapter 18.

Topological combinatorics The use of topology, such as the Borsuk-Ulam theorem, to prove combinatorial statements; Lovász started it with his proof of Kneser's conjecture. See Chapter 17.

Topological data analysis The study of the shape of data, such as point clouds, with tools from algebraic topology, chiefly persistent homology. See Chapter 17.

Topological insulator A material that insulates in its interior but conducts along its surface, protected by a topological invariant of its electron states. See Chapter 20.

Topological invariant A property or quantity that homeomorphic spaces share. Connectedness, compactness, the Euler characteristic, the fundamental group and homology are examples. See Chapter 1 and Chapter 5.

Topological phase transition A change of state of matter marked by a change of a topological invariant, such as the BKT transition, in which vortices unbind. See Chapter 20.

Topological property A property that survives every homeomorphism, the subject of topology. See Chapter 1.

Topological quantum computing A proposed way to compute by braiding non-abelian anyons, so that the result depends only on the topology of the braid and resists small errors. See Chapter 20.

Topological quantum field theory A physical theory whose answers depend only on the topology of space-time; Witten showed the Jones polynomial arises from one. See Chapter 20.

Topological realisation The space \(|G|\) of a graph: a point for each vertex and an arc for each edge, glued at the endpoints. See Chapter 3.

Topological space A set \(X\) with a topology: a collection of subsets, the open sets, that contains \(\varnothing\) and \(X\) and is closed under arbitrary unions and finite intersections. See Chapter 1 and Chapter 4.

Topologist's sine curve The graph of \(\sin(1/x)\) for \(0 < x \le 1\) together with the segment of the \(y\)-axis it accumulates on. It is connected but not path connected. See Chapter 7.

Topology (on a set) A collection of subsets, called open, containing the empty set and the whole set and closed under arbitrary unions and finite intersections. See Chapter 4.

Torsion The elements of finite order in a finitely generated abelian group, such as the \(\mathbb{Z}/2\) in \(H_1(\mathbb{RP}^2)\). Betti numbers ignore it. See Chapter 16.

Torus The surface of a doughnut, \(S^1 \times S^1\), or a square with opposite sides glued the same way (word \(aba^{-1}b^{-1}\)). \(\chi = 0\) and \(\pi_1 = \mathbb{Z}^2\). The \(n\)-torus is the product of \(n\) circles. See Chapter 1, Chapter 2, Chapter 6 and Chapter 15.

Totally bounded A metric space that can be covered by finitely many balls of radius \(\varepsilon\) for every \(\varepsilon > 0\). A metric space is compact exactly when it is complete and totally bounded. See Chapter 8.

Totally disconnected A space whose only connected subsets are single points, such as \(\mathbb{Q}\) or the Cantor set. See Chapter 7 and Chapter 9.

Transition map For two overlapping charts \(\varphi\) and \(\psi\) of a manifold, the map \(\psi \circ \varphi^{-1}\) between open sets of \(\mathbb{R}^n\). See Chapter 11.

Tree A connected graph with no cycles. A tree with \(V\) vertices has \(V - 1\) edges, and it is contractible. See Chapter 2 and Chapter 12.

Trefoil The simplest non-trivial knot, with three crossings. It is tricolourable and chiral, and its knot group is the braid group on three strands. See Chapter 15 and Chapter 18.

Triangle inequality The axiom \(d(x, z) \le d(x, y) + d(y, z)\) of a metric: a detour is never shorter. See Chapter 4.

Triangulation A homeomorphism from a polyhedron \(|K|\) of a simplicial complex to a space. Every surface and every 3-manifold has one. See Chapter 16.

Triangulation conjecture The question whether every topological manifold is homeomorphic to a simplicial complex. Manolescu proved it false in every dimension \(n \ge 5\); it was already false in dimension 4. See Chapter 20.

Tricolouring A colouring of the arcs of a knot diagram with three colours, at each crossing all the same or all different, using more than one colour. A knot with one is tricolourable, which proves the trefoil knotted. See Chapter 15 and Chapter 18.

Trivial group The group with one element. A space is simply connected when its fundamental group is trivial. See Chapter 13.

Tube lemma If \(C\) is compact and \(N\) is an open set containing \(\{x\} \times C\) in \(X \times C\), then \(N\) contains a tube \(W \times C\) around it. It proves that finite products of compact spaces are compact. See Chapter 8.

Tychonoff's theorem Any product of compact spaces is compact in the product topology. For infinite products it is equivalent to the axiom of choice. See Chapter 8.

U

Undecidable A problem for which no algorithm gives the correct answer on every input. The word problem for groups, and the homeomorphism problem for 4-manifolds, are undecidable. See Chapter 15 and Chapter 19.

Universal cover A simply connected covering space of a space. It covers every other connected covering, and its deck transformation group is the fundamental group. See Chapter 14.

Unknot A knot equivalent to a round circle. Haken's algorithm decides whether a diagram shows the unknot. See Chapter 15, Chapter 18 and Chapter 18.

Urysohn function For disjoint closed sets \(A\) and \(B\), a continuous function \(f: X \to [0, 1]\) equal to 0 on \(A\) and 1 on \(B\). See Chapter 9.

Urysohn's lemma In a normal space, any two disjoint closed sets have an Urysohn function: they are separated by a continuous function. See Chapter 9.

Urysohn's metrisation theorem Every regular, second countable space is metrisable; it embeds in the Hilbert cube. See Chapter 9.

V

Vertex A point of a graph where edges end, or a corner of a polyhedron or of a labelled polygon. See Chapter 1, Chapter 2 and Chapter 10.

Vietoris-Rips complex The simplicial complex built on a point cloud at scale \(r\), with a simplex on every set of points that are pairwise within distance \(r\). See Chapter 17.

Volume conjecture The open conjecture that the growth of coloured Jones polynomials of a knot determines the hyperbolic volume of its complement. See Chapter 20.

Von Dyck's theorem A function from the generators \(S\) to a group \(G\) extends to a homomorphism \(\langle S \mid R \rangle \to G\) exactly when every relator goes to 1, and the extension is then unique. See Chapter 15.

Vortex A point in a two-dimensional field around which the field's direction winds a nonzero number of times. Vortices unbinding drives the BKT transition. See Chapter 20.

W

Wagner's theorem A graph is planar exactly when it has neither \(K_5\) nor \(K_{3,3}\) as a minor. See Chapter 3.

Wedge sum The space \(X \vee Y\) made by gluing \(X\) and \(Y\) at one point. The wedge of two circles is the figure eight. See Chapter 6 and Chapter 15.

Whitehead manifold An open contractible 3-manifold not homeomorphic to \(\mathbb{R}^3\), found by J. H. C. Whitehead in 1935 while correcting his false proof of the Poincaré conjecture. See Chapter 19.

Whitehead's asphericity conjecture The open conjecture (1941) that every subcomplex of an aspherical 2-dimensional complex is aspherical. See Chapter 20.

Whitney embedding theorem Every smooth \(n\)-manifold embeds smoothly in \(\mathbb{R}^{2n}\). See Chapter 11.

Whitney trick A move that removes a pair of opposite intersection points of two submanifolds by pushing one across a Whitney disc. It works in dimensions 5 and up, and fails in dimension 4. See Chapter 19.

Winding number The net number of times a loop goes around a point, anticlockwise counted positive. It identifies \(\pi_1(S^1)\) with \(\mathbb{Z}\). See Chapter 13 and Chapter 14.

Word (in a group) A finite sequence of letters and their inverses, such as \(abb^{-1}a^{-1}a\). Reduced words are the elements of a free group. See Chapter 15.

Word (of a labelled polygon) The sequence of edge letters read around a polygon, with exponent \(-1\) when an arrow points against the direction of travel. It records how the polygon is glued into a surface. See Chapter 2 and Chapter 10.

Word problem The problem of deciding, from a presentation of a group, whether a word represents the identity. Novikov and Boone showed it is unsolvable in general; Dehn's algorithm solves it for surfaces. See Chapter 15.

Writhe The sum of the signs of all crossings of an oriented knot diagram. In DNA, the writhe measures how the axis coils on itself. See Chapter 18 and Chapter 18.

Z

Zariski topology The topology on \(k^n\) whose closed sets are the common zero sets of polynomials. It is T1 but not Hausdorff. See Chapter 4 and Chapter 9.

ZIP proof Conway's "zero irrelevancy proof" of the classification of surfaces. See Chapter 10.