20. Where topology goes¶

You started this book with a coffee cup and a doughnut and ended Part V with the Poincaré conjecture. Topology does not stop there. This last chapter is a guided tour of the frontier: spheres that are the same shape but not the same smooth shape, a version of Poincaré's question in four dimensions that nobody can answer, the homotopy groups of spheres that nobody can fully compute, topology inside physics, where integers measured in a laboratory turn out to be topological invariants, the idea of computing with braids, and a list of open problems. Each section states things honestly, proves what can be proved in a page, gives examples you can check, and says where the full story is told.

In this chapter

You will learn:

what a smooth structure is, Milnor's exotic 7-spheres, and the strange case of exotic \(\mathbb{R}^4\)
the smooth 4-dimensional Poincaré conjecture, the main open problem of its kind
the homotopy groups of spheres, the Hopf map \(S^3 \to S^2\), and why no one has a formula for them
how topology explains quantised quantities in physics, and what the 2016 Nobel Prize in Physics was for
the idea of topological quantum computing, and its honest status
a list of open problems, and some recently solved ones

You need: the whole book, especially Chapter 11 (manifolds), Chapter 14 (winding numbers), Chapter 17 (degree), Chapter 18 (knots, the Jones polynomial, braids) and Chapter 19 (the Poincaré conjecture and its higher-dimensional versions). This chapter states more than it proves; where a proof is given it is complete, and where it is not, the text says where to find it.

Levels: all sections are expert in content but written as a survey. 20.3 and 20.6 can be read after Part IV; 20.1 and 20.2 build on Chapter 19; 20.4 and 20.5 need some physics vocabulary, explained as it comes.

20.1 Exotic spheres and exotic \(\mathbb{R}^4\)¶

Level: expert. In Chapter 11 a smooth structure on a manifold was an atlas of charts whose transition maps are infinitely differentiable. Two smooth manifolds are diffeomorphic if there is a homeomorphism between them that is smooth with smooth inverse. A homeomorphism can have corners and kinks; a diffeomorphism cannot. For a long time it seemed obvious that a manifold homeomorphic to a sphere would be diffeomorphic to it. It is not true.

Milnor's spheres¶

In 1956 John Milnor constructed smooth 7-dimensional manifolds that are homeomorphic to \(S^7\) but not diffeomorphic to it. He built them as bundles over \(S^4\) with fibre \(S^3\) (each looks locally like a piece of \(S^4\) times a 3-sphere, glued with a twist given by unit quaternions), showed by a Morse function with only two critical points that some of them are homeomorphic to \(S^7\) (a manifold with such a function is a union of two discs glued along the boundary, as in 19.2), and found an invariant, computed from the characteristic classes of a bounding 8-manifold, that is defined only for smooth structures and differs from its value on the standard \(S^7\). Such manifolds are called exotic spheres. Milnor received a Fields Medal in 1962.

In 1963 Michel Kervaire and Milnor organised all of them. For \(n \ge 5\) the smooth homotopy \(n\)-spheres, up to orientation-preserving diffeomorphism, form a finite abelian group \(\Theta_n\) under connected sum, with the standard sphere as identity. The orders of the first groups are

\(n\)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
\(\lvert\Theta_n\rvert\)	1	1	1	?	1	1	28	2	8	6	992	1	3	2	16256

So there are 28 oriented smooth structures on the 7-sphere, and 992 on the 11-sphere. (In dimensions 1 and 2 uniqueness is classical, and in dimension 3 it follows from Moise's theorem that every topological 3-manifold has a unique smooth structure, together with Perelman. Dimension 4 is the subject of 20.2.) In 1966 Egbert Brieskorn found the 28 exotic 7-spheres in algebraic geometry: the intersection of the complex hypersurface \(z_1^{6k-1} + z_2^3 + z_3^2 + z_4^2 + z_5^2 = 0\) in \(\mathbb{C}^5\) with a small sphere \(S^9\) around the origin is, for \(k = 1, 2, \dots, 28\), each of the 28 in turn.

The computation of \(\Theta_n\) reduces, by Kervaire and Milnor's work, to the stable homotopy groups of spheres (20.3), which is why it is not known in general. In 2017 Guozhen Wang and Zhouli Xu completed the proof that in odd dimensions the only spheres with a unique smooth structure are \(S^1\), \(S^3\), \(S^5\) and \(S^{61}\).

Exotic \(\mathbb{R}^4\)¶

For Euclidean space the answer is stranger. \(\mathbb{R}^n\) has exactly one smooth structure up to diffeomorphism for every \(n \ne 4\) (for \(n \ge 5\) this is a theorem of Stallings from 1962; for \(n \le 3\) it is classical). For \(n = 4\) there are exotic \(\mathbb{R}^4\)s: smooth manifolds homeomorphic to \(\mathbb{R}^4\) but not diffeomorphic to it. They came out of two results of the early 1980s that pull in opposite directions:

Freedman's work (19.2) showed that topological 4-manifolds are flexible: for instance, there is a closed topological 4-manifold whose intersection form (the pairing of 2-dimensional homology classes) is the \(E_8\) form.
Simon Donaldson's theorem (1983), proved with gauge theory from physics (the Yang-Mills equations), showed that smooth 4-manifolds are rigid: a closed, simply connected smooth 4-manifold whose intersection form is definite must have a diagonalisable form. The \(E_8\) manifold therefore has no smooth structure.

Combining the two produces exotic smooth structures on \(\mathbb{R}^4\), and Clifford Taubes showed in 1987 that there are uncountably many. Donaldson and Freedman both received Fields Medals in 1986. Four is the only dimension in which Euclidean space is not unique, and nobody has a short explanation of why four is so special, beyond the observation that it is high enough for complicated phenomena and too low for the Whitney trick.

20.2 The smooth 4-dimensional Poincaré conjecture¶

Level: expert. Freedman proved that a closed 4-manifold homotopy equivalent to \(S^4\) is homeomorphic to \(S^4\). The smooth version is open:

Smooth 4-dimensional Poincaré conjecture. A smooth closed 4-manifold homotopy equivalent to \(S^4\) is diffeomorphic to \(S^4\).

It is the last unsolved case of the generalised Poincaré conjecture in the smooth category below dimension 5 (in higher dimensions the smooth version is false, by 20.1). Experts disagree about whether it is true. Some reasons for the difficulty:

The tools that classify topological 4-manifolds (Freedman) cannot see smooth structures, and the tools that see smooth structures (Donaldson's invariants, and Seiberg-Witten invariants from 1994) give no information on homotopy 4-spheres: they vanish or are undefined there.
Several families of proposed counterexamples, built by cutting and regluing (Gluck twists, and the spheres of Cappell and Shaneson from 1976), have turned out, one after another, to be standard after all, sometimes after decades and long explicit sequences of handle moves.
A related question is the smooth 4-dimensional Schoenflies problem: does every smoothly embedded 3-sphere in \(S^4\) bound a smooth ball on each side? It is also open.

New invariants from knot theory have been tried. If a knot in \(S^3\) bounds a smooth disc in some homotopy 4-ball but not in the standard 4-ball, that would disprove the conjecture; Freedman, Gompf, Morrison and Walker proposed using Khovanov homology (18.3) to detect this, but the candidates they examined did not give a counterexample. Related techniques have had other successes: in 2020 Lisa Piccirillo proved that the Conway knot, an 11-crossing knot whose status had been open for decades, is not slice (does not bound a smooth disc in the 4-ball).

20.3 Homotopy groups of spheres¶

Level: expert. The fundamental group uses loops, maps \(S^1 \to X\). Use spheres of higher dimension and you get the higher homotopy groups:

\[ \pi_k(X) = \{\text{maps } S^k \to X \text{ sending a base point to a base point}\} / \text{homotopy} . \]

Composition is defined as for loops, by squeezing the equator of \(S^k\) to a point and using one map on each half. For \(k \ge 2\) the group is abelian: in a \(k\)-dimensional cube there is room to slide the two halves around each other. Homotopy groups are the most natural invariants imaginable, and computing them for the simplest spaces, the spheres, is one of the hardest problems in mathematics.

What is known easily¶

\(\pi_k(S^n) = 0\) for \(k < n\). A map \(S^k \to S^n\) can be deformed to a smooth one, whose image has dimension at most \(k < n\) and so misses a point; the sphere minus a point is contractible, so the map is null-homotopic. (This is the argument by which \(\pi_1(S^2) = 0\) in Chapter 13.)
\(\pi_n(S^n) \cong \mathbb{Z}\), the isomorphism being the degree (Chapter 17, Hopf's theorem).
\(\pi_k(S^1) = 0\) for \(k \ge 2\), because the universal cover of \(S^1\) is the contractible line (Chapter 14) and covering maps induce isomorphisms on \(\pi_k\) for \(k \ge 2\).

So for the circle everything is known. For \(S^2\) the next group already holds a surprise.

The Hopf map: \(\pi_3(S^2) \cong \mathbb{Z}\)¶

Homology says \(S^2\) has nothing in dimension 3: \(H_3(S^2) = 0\). Yet in 1931 Heinz Hopf found a map \(S^3 \to S^2\) that is not null-homotopic. Think of \(S^3\) as pairs of complex numbers \((z, w)\) with \(|z|^2 + |w|^2 = 1\), and \(S^2\) as the unit sphere in \(\mathbb{R}^3 = \mathbb{C} \times \mathbb{R}\). The Hopf map is

\[ H(z, w) = \left(2 z \bar w,\; |z|^2 - |w|^2\right) . \]

Worked example 1. It lands on \(S^2\):

\[ |2z\bar w|^2 + \left(|z|^2 - |w|^2\right)^2 = 4|z|^2|w|^2 + |z|^4 - 2|z|^2|w|^2 + |w|^4 = \left(|z|^2 + |w|^2\right)^2 = 1 . \]

Its fibres are circles: multiplying both coordinates by \(e^{i\theta}\) does not change \(z \bar w\) (the factors \(e^{i\theta}\) and \(e^{-i\theta}\) cancel) or \(|z|, |w|\), so \(H\) is constant on each circle \(\{(e^{i\theta}z, e^{i\theta}w)\}\). Conversely, \(H(z,w)\) determines \((z, w)\) up to such a factor. So \(S^3\) is a union of disjoint circles, one over each point of \(S^2\): the Hopf fibration.

Two fibres: over the north pole \((0, 1)\) we need \(|w| = 0\), giving the circle \(\{(e^{i\theta}, 0)\}\); over the south pole \((0, -1)\) we need \(z = 0\), giving \(\{(0, e^{i\theta})\}\). These two great circles of \(S^3\), seen through stereographic projection into \(\mathbb{R}^3\), become a round circle and a straight line through it (closing up through the point at infinity): they are linked once, a Hopf link (Chapter 18). In fact any two fibres are linked once.

The Hopf invariant of a smooth map \(f: S^3 \to S^2\) is the linking number of the preimages of two different regular values. It is a homotopy invariant (Hatcher, section 4.B, or Bott and Tu's Differential Forms in Algebraic Topology). The constant map has Hopf invariant 0 and \(H\) has Hopf invariant 1, so \(H\) is not null-homotopic, and in fact \(\pi_3(S^2) \cong \mathbb{Z}\), generated by \(H\), with the Hopf invariant as the isomorphism.

Try it in Python. Check both facts at random points:

import cmath, math, random

def hopf(z, w):
    u = 2 * z * w.conjugate()
    return (u.real, u.imag, abs(z)**2 - abs(w)**2)

for _ in range(3):
    v = [random.gauss(0, 1) for _ in range(4)]
    r = math.sqrt(sum(x * x for x in v))
    z, w = complex(v[0], v[1]) / r, complex(v[2], v[3]) / r
    p = hopf(z, w)
    q = hopf(cmath.exp(1j) * z, cmath.exp(1j) * w)       # another point of the same circle
    print(round(sum(x * x for x in p), 12), max(abs(a - b) for a, b in zip(p, q)) < 1e-12)
# 1.0 True   (three times)

The table, and why there is no formula¶

Here are some of the groups, with \(\mathbb{Z}_m\) meaning the cyclic group of order \(m\).

	\(\pi_1\)	\(\pi_2\)	\(\pi_3\)	\(\pi_4\)	\(\pi_5\)	\(\pi_6\)	\(\pi_7\)
\(S^1\)	\(\mathbb{Z}\)	0	0	0	0	0	0
\(S^2\)	0	\(\mathbb{Z}\)	\(\mathbb{Z}\)	\(\mathbb{Z}_2\)	\(\mathbb{Z}_2\)	\(\mathbb{Z}_{12}\)	\(\mathbb{Z}_2\)
\(S^3\)	0	0	\(\mathbb{Z}\)	\(\mathbb{Z}_2\)	\(\mathbb{Z}_2\)	\(\mathbb{Z}_{12}\)	\(\mathbb{Z}_2\)
\(S^4\)	0	0	0	\(\mathbb{Z}\)	\(\mathbb{Z}_2\)	\(\mathbb{Z}_2\)	\(\mathbb{Z} \oplus \mathbb{Z}_{12}\)

Some patterns can be proved:

Serre (1951). \(\pi_k(S^n)\) is finite except for \(\pi_n(S^n) \cong \mathbb{Z}\) and, for even \(n\), \(\pi_{2n-1}(S^n)\), which is \(\mathbb{Z}\) plus a finite group. (Look at \(\pi_3(S^2)\) and \(\pi_7(S^4)\) in the table.)
Freudenthal (1937). Suspension (Chapter 6) gives maps \(\pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})\) that are isomorphisms once \(n > k + 1\). So along each diagonal of the table the groups eventually stop changing. The limits are the stable homotopy groups of spheres \(\pi_k^s\):

\(k\)	0	1	2	3	4	5	6	7
\(\pi_k^s\)	\(\mathbb{Z}\)	\(\mathbb{Z}_2\)	\(\mathbb{Z}_2\)	\(\mathbb{Z}_{24}\)	0	0	\(\mathbb{Z}_2\)	\(\mathbb{Z}_{240}\)

The 24 and the 240 are not random: they come from Bernoulli numbers, through the "image of J", a part of the stable groups that Frank Adams computed in the 1960s.

Adams (1960) settled the Hopf invariant one problem: maps \(S^{2n-1} \to S^n\) of Hopf invariant one exist only for \(n = 2, 4, 8\) (built from the complex numbers, quaternions and octonions, as \(H\) is built from \(\mathbb{C}\)). A corollary: \(\mathbb{R}^n\) can be given a continuous multiplication with a two-sided identity and no zero divisors only for \(n = 1, 2, 4, 8\). That real division algebras exist only in these dimensions had been proved in 1958 by Bott and Milnor and by Kervaire, also with topology.

Beyond such patterns, the groups are computed one at a time with heavy machinery (spectral sequences, and lately computer calculation). The stable groups are now known through dimension around 90 (Isaksen, Wang and Xu, 2023). There is no formula, and many topologists doubt that a closed formula exists. The Kervaire invariant problem, about certain elements of these groups, was resolved in all dimensions except 126 by Hill, Hopkins and Ravenel (published 2016); that last case has been the subject of more recent work.

20.4 Topology in physics¶

Level: expert (survey). Topological invariants are integers, and integers cannot change a little. If a physical quantity is secretly a topological invariant, it will be exactly quantised and immune to small imperfections. That is the theme of this section.

A model: Gauss-Bonnet. For a closed surface with any Riemannian metric, the total curvature is \(2\pi\) times an integer:

\[ \int_\Sigma K \, dA = 2\pi\, \chi(\Sigma) . \]

Dent the surface and \(K\) changes everywhere, but the integral does not move. Check on a round sphere of radius \(r\): \(K = 1/r^2\), area \(4\pi r^2\), integral \(4\pi = 2\pi \cdot 2\).

The quantum Hall effect. In 1980 Klaus von Klitzing found that a thin layer of electrons at low temperature in a strong magnetic field has a Hall conductance (the sideways current per unit voltage) that is exactly an integer multiple of \(e^2/h\), where \(e\) is the electron charge and \(h\) is Planck's constant, to a precision of parts per billion, no matter how impure the sample. (He received the 1985 Nobel Prize in Physics.) In 1982 David Thouless, Mahito Kohmoto, Peter Nightingale and Marcel den Nijs explained the integer: it is the Chern number of a complex line bundle over a torus (the "Brillouin zone" of electron momenta, which is a torus because momenta in a crystal are periodic). The Chern number is the integral of a curvature, divided by \(2\pi\), just like Gauss-Bonnet, and it is an integer for the same reason.

Try it in Python. A standard toy model has, at each momentum \((k_x, k_y)\) of the torus, a \(2 \times 2\) matrix whose lower eigenvector (the occupied state) we follow around the torus. The Chern number counts how many times that eigenvector "winds" as you cover the torus; a lattice method (Fukui, Hatsugai and Suzuki, 2005) computes it by adding the phase changes around each small square.

import cmath, math

def lower_band(kx, ky, m):
    dx, dy, dz = math.sin(kx), math.sin(ky), m + math.cos(kx) + math.cos(ky)
    e = math.sqrt(dx * dx + dy * dy + dz * dz)
    u, v = complex(dx, -dy), -(dz + e)
    if abs(u) + abs(v) < 1e-9:                  # use the other chart
        u, v = complex(e - dz), -complex(dx, dy)
    n = math.sqrt(abs(u) ** 2 + abs(v) ** 2)
    return u / n, v / n

def chern(m, N=40):
    k = [2 * math.pi * i / N for i in range(N)]
    s = [[lower_band(a, b, m) for b in k] for a in k]
    dot = lambda p, q: p[0].conjugate() * q[0] + p[1].conjugate() * q[1]
    total = 0.0
    for i in range(N):
        for j in range(N):
            a, b = s[i][j], s[(i + 1) % N][j]
            c, d = s[(i + 1) % N][(j + 1) % N], s[i][(j + 1) % N]
            total += cmath.phase(dot(a, b) * dot(b, c) * dot(c, d) * dot(d, a))
    return round(total / (2 * math.pi))

for m in [-3, -1, 1, 3]:
    print(m, chern(m))      # -3 0, -1 -1, 1 1, 3 0

The answer is \(+1\) for \(0 < m < 2\), \(-1\) for \(-2 < m < 0\), and \(0\) for \(|m| > 2\). It can change only at \(m = 0, \pm 2\), where the two eigenvalues \(\pm|d|\) meet (the vector \(d\) vanishes at some momentum). That is a topological phase transition: the integer cannot change smoothly, so it changes only where the structure breaks down.

Vortices and the BKT transition. In a thin film of a superfluid, or in a two-dimensional model of magnetic arrows free to turn in a plane, the defects are vortices: points around which the arrows wind once, as a winding number in the sense of Chapter 14. In the early 1970s Vadim Berezinskii, and independently J. Michael Kosterlitz and David Thouless, showed that such systems have a phase transition of a new kind: at low temperature vortices exist only in tightly bound pairs of opposite winding, and above a critical temperature they unbind.

Topological phases. In the 1980s Duncan Haldane found topological effects in chains of magnetic atoms (1983), and a model (1988) showing that the quantum Hall effect does not need an overall magnetic field. Later, Charles Kane and Eugene Mele (2005) predicted topological insulators, materials that insulate inside but must conduct along their edges or surfaces because of a topological invariant of their bands; the first was observed in 2007.

The 2016 Nobel Prize in Physics went one half to David Thouless and the other half to Duncan Haldane and Michael Kosterlitz, "for theoretical discoveries of topological phase transitions and topological phases of matter".

Two honest remarks. First, the topology here is usually of an abstract space, such as a bundle over the torus of momenta, not the shape of the material. Second, the mathematics needed (vector bundles, characteristic classes, K-theory for the classification of topological insulators) goes beyond this book; it builds on the homology of Chapter 16 and the covering spaces of Chapter 14.

Topological quantum field theory. In 1988 Michael Atiyah wrote down axioms for a topological quantum field theory: a rule that assigns vector spaces to surfaces and linear maps to 3-manifolds bounded by them, compatible with gluing. In 1989 Edward Witten showed that Chern-Simons theory, a quantum field theory in three dimensions, gives such a rule and that its values on knots are the Jones polynomial of Chapter 18. Witten received a Fields Medal in 1990, the same year as Vaughan Jones.

20.5 Topological quantum computing¶

Level: expert (survey, brief). A quantum computer stores information in fragile quantum states, which noise destroys. The idea of topological quantum computing, due mainly to Alexei Kitaev (around 1997, published 2003) and Michael Freedman, is to store information in topological properties of a physical system, which local noise cannot change, as local dents cannot change the integral in Gauss-Bonnet.

Braids instead of swaps. In three-dimensional space, swapping two identical particles twice is the same as doing nothing: the path can be deformed away. In two dimensions it cannot, because the paths of the particles in spacetime form a braid, and braids can be tangled (Chapter 18 used braids to draw the trefoil). The braid group \(B_n\) on \(n\) strands has generators \(\sigma_1, \dots, \sigma_{n-1}\) (strand \(i\) crosses over strand \(i+1\)) and relations

\[ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \qquad \sigma_i \sigma_j = \sigma_j \sigma_i \text{ for } |i - j| \ge 2 . \]

The first relation is the R3 move of Chapter 18. Particles in two dimensions whose exchanges act on the quantum state by matrices depending on the braid are called anyons; if the matrices do not commute they are non-abelian anyons. A computation is then a braid: move the anyons around each other, and the result depends only on the braid's topology. Freedman, Larsen and Wang showed (2002) that certain anyon models, such as "Fibonacci anyons", can approximate any quantum computation. Conversely, a quantum computer can approximate values of the Jones polynomial at certain roots of unity efficiently (Aharonov, Jones and Landau, 2006), a task believed hard for ordinary computers.

Kitaev's toric code. In Kitaev's model, qubits sit on the edges of a grid drawn on a surface, and the protected states are counted by topology: on a closed orientable surface of genus \(g\) there are \(4^g\) of them, so 4 on a torus. Errors that act locally cannot move between them; only an error that runs all the way around a non-contractible loop can.

Status, honestly. Non-abelian anyons are predicted in certain quantum Hall states and in engineered superconducting devices hosting "Majorana zero modes". Experimental claims have been made and disputed; one prominent 2018 claim of a Majorana signature was retracted in 2021. Whether a useful topological qubit can be built is an open experimental question, and readers should check current literature rather than any book, including this one.

20.6 Open problems¶

Level: expert (survey). Some problems from this book that are still open, stated as precisely as a survey allows:

Smooth 4D Poincaré conjecture (20.2) and the smooth 4D Schoenflies problem.
Homotopy groups of spheres (20.3). No general formula; each new range of the stable groups is a research project.
Unknot recognition (18.5). Is there a polynomial-time algorithm?
Does the Jones polynomial detect the unknot? (18.3). Is \(V_K = 1\) only for the unknot?
The slice-ribbon conjecture (Ralph Fox, 1960s). Every knot that bounds a smooth disc in the 4-ball bounds one of a special kind, a "ribbon" disc that can be pushed into the 3-sphere with only simple self-intersections.
The volume conjecture (Kashaev; H. Murakami and J. Murakami, around 2000). The growth of certain "coloured" Jones polynomials of a knot determines the hyperbolic volume of its complement, linking Chapter 18 to the hyperbolic geometry of Chapter 19.
The Andrews-Curtis conjecture (1965). Every balanced presentation of the trivial group can be reduced to the trivial presentation by a specific list of moves. Many believe it false, but no counterexample has been proved.
Whitehead's asphericity conjecture (1941). Every subcomplex of an aspherical 2-dimensional complex (one whose universal cover is contractible) is aspherical.
The Borel conjecture. Two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic. This is a rigidity statement in the spirit of Mostow's theorem (Chapter 19), believed but proved only in many special cases.

And some that were open and are now settled, to show that the frontier moves:

Poincaré in dimension 3 (Perelman, 2002 to 2003) and geometrisation (Chapter 19).
Virtual Haken conjecture (Agol, 2012): every closed hyperbolic 3-manifold has a finite cover containing an incompressible surface (19.5).
Triangulation conjecture (is every topological manifold homeomorphic to a simplicial complex?): false in every dimension \(n \ge 5\), by Ciprian Manolescu (published 2016), building on earlier work that reduced it to a question about homology 3-spheres like Poincaré's (19.1). In dimension 4 it was already known to be false, by Freedman's \(E_8\) manifold and work of Casson.
The Conway knot is not slice (Piccirillo, 2020; 20.2).
Hilbert-Smith conjecture in dimension 3 (Pardon, 2013): no \(p\)-adic group acts faithfully on a 3-manifold.

History thread: the whole book¶

Euler solved the Königsberg bridges problem in 1736 (Chapter 1) and found \(V - E + F = 2\) (Chapter 2). Riemann, in the 1850s, used the number of cuts that leave a surface connected, which is the genus, to study functions of a complex variable. Poincaré, from 1895, created homology and the fundamental group and asked his question. Brouwer, around 1910 to 1912, proved invariance of dimension and his fixed point theorem. The point-set axioms took their modern form with Hausdorff (1914) and others; Alexander, Lefschetz, Hopf and others built algebraic topology between the world wars; Smale, Milnor, Thurston, Freedman and Donaldson transformed the topology of manifolds in the second half of the twentieth century; Perelman closed Poincaré's question at its start. The field keeps growing, now with computers, data and physics among its customers.

Check yourself¶

1. Show that \(H(z, w) = (2z\bar w, |z|^2 - |w|^2)\) lands on the unit sphere when \(|z|^2 + |w|^2 = 1\).

\(|2z\bar w|^2 = 4|z|^2|w|^2\), and \((|z|^2 - |w|^2)^2 = |z|^4 - 2|z|^2|w|^2 + |w|^4\). Adding gives \(|z|^4 + 2|z|^2|w|^2 + |w|^4 = (|z|^2 + |w|^2)^2 = 1\).

2. Find the fibre of the Hopf map over the point \((1, 0, 0)\), that is \(2z\bar w = 1\) and \(|z|^2 = |w|^2\).

\(|z|^2 = |w|^2 = \tfrac12\), and \(z\bar w = \tfrac12\) is real and positive, so \(z\) and \(w\) have the same argument: \(z = w\). The fibre is \(\{(e^{i\theta}/\sqrt2, e^{i\theta}/\sqrt2)\}\), a great circle. Check with \(\theta = 0\): \(2 \cdot \tfrac{1}{\sqrt2}\cdot\tfrac{1}{\sqrt2} = 1\) and \(\tfrac12 - \tfrac12 = 0\).

3. Why is \(\pi_3(S^2) \ne 0\) not a contradiction with \(H_3(S^2) = 0\)?

Homotopy and homology are different invariants. The Hurewicz theorem (Chapter 16) identifies them only in the first dimension where a simply connected space has non-trivial homotopy: for \(S^2\), \(\pi_2 \cong H_2 \cong \mathbb{Z}\). Above that there is no such relation, and \(\pi_3(S^2) \cong \mathbb{Z}\) while \(H_3(S^2) = 0\).

4. \(\Theta_7 \cong \mathbb{Z}/28\) counts exotic 7-spheres up to orientation-preserving diffeomorphism. Reversing orientation sends a sphere to its inverse in the group. How many classes are there if orientation is ignored?

Pair each element \(k\) with \(-k\) in \(\mathbb{Z}/28\). The elements with \(k = -k\) are \(0\) and \(14\). The other 26 form 13 pairs. So there are \(2 + 13 = 15\) smooth 7-spheres up to diffeomorphism, ignoring orientation (one standard and 14 exotic).

5. Use Freudenthal's theorem to explain why \(\pi_4(S^3) \cong \pi_5(S^4) \cong \mathbb{Z}_2\) but \(\pi_3(S^2) \not\cong \pi_4(S^3)\).

Here \(k = 1\), and the suspension \(\pi_{n+1}(S^n) \to \pi_{n+2}(S^{n+1})\) is an isomorphism for \(n > 2\). So from \(n = 3\) on, the group is stable: \(\pi_4(S^3) \cong \pi_5(S^4) \cong \pi_1^s = \mathbb{Z}_2\). For \(n = 2\) the theorem only gives a surjection \(\pi_3(S^2) = \mathbb{Z} \to \pi_4(S^3) = \mathbb{Z}_2\), which is reduction mod 2.

6. Using Gauss-Bonnet, what is the total curvature of a torus with any metric, and of a surface of genus 3?

\(\chi(T^2) = 0\), so \(\int K\,dA = 0\): the positive curvature on the outside exactly cancels the negative curvature on the inside. For genus 3, \(\chi = 2 - 6 = -4\), so the total curvature is \(-8\pi\).

7. In the Python model, find the momenta where the vector \(d = (\sin k_x, \sin k_y, m + \cos k_x + \cos k_y)\) vanishes when \(m = -2\), \(m = 0\) and \(m = 2\).

\(d = 0\) needs \(\sin k_x = \sin k_y = 0\), so \(k_x, k_y \in \{0, \pi\}\), and then \(m + \cos k_x + \cos k_y = 0\). The sums \(\cos k_x + \cos k_y\) are \(2\) at \((0,0)\), \(0\) at \((0,\pi)\) and \((\pi, 0)\), and \(-2\) at \((\pi,\pi)\). So: \(m = -2\) at \((0,0)\); \(m = 0\) at \((0,\pi)\) and \((\pi,0)\); \(m = 2\) at \((\pi,\pi)\). These are the only values where the Chern number can change.

8. Check that the braid relation \(\sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2\) holds in the symmetric group when \(\sigma_1 \mapsto (1\,2)\) and \(\sigma_2 \mapsto (2\,3)\). Why does \(\sigma_1^2 = 1\) hold there but not in the braid group?

Composing right to left, \((1\,2)(2\,3)(1\,2)\) sends \(1 \to 2 \to 3 \to 3\), \(3 \to 3 \to 2 \to 1\), \(2 \to 1 \to 1 \to 2\): it is \((1\,3)\). Similarly \((2\,3)(1\,2)(2\,3) = (1\,3)\). A transposition squared is the identity, but \(\sigma_1^2\) is a full twist of two strands, which is not the trivial braid: its closure is the Hopf link (linking number 1), while the closure of the trivial 2-strand braid is the two-component unlink.

9. How many protected states does Kitaev's toric code have on a surface of genus 2? Why does the answer depend only on the genus?

\(4^2 = 16\). The protected states are labelled by what happens around non-contractible loops, and on a genus-\(g\) surface \(H_1 \cong \mathbb{Z}^{2g}\); with \(\mathbb{Z}/2\) coefficients that is \(2g\) independent loops, each giving a factor of 2, for \(2^{2g} = 4^g\) states. A topological invariant of the surface fixes the count.

10. Which real vector spaces \(\mathbb{R}^n\) admit a bilinear multiplication without zero divisors, and which known algebras realise them?

Only \(n = 1, 2, 4, 8\): the real numbers, the complex numbers, the quaternions and the octonions. That nothing else exists was proved with topology (Bott-Milnor and Kervaire, 1958), and Adams's Hopf invariant one theorem gives a stronger form.

11. Why does the existence of an exotic \(\mathbb{R}^4\) not contradict Freedman's result that every homotopy 4-sphere is homeomorphic to \(S^4\)?

Freedman's result is about homeomorphism. An exotic \(\mathbb{R}^4\) is homeomorphic to \(\mathbb{R}^4\); it only fails to be diffeomorphic. Topological and smooth classifications differ in dimension 4, which is why the smooth 4-dimensional Poincaré conjecture is still open while the topological one is proved.

12. Give one reason each for: (a) why topological quantities in physics are measured so precisely; (b) why that alone does not guarantee a working topological quantum computer.

(a) A topological invariant is an integer that does not change under small perturbations, so impurities and noise cannot shift it a little; the measured Hall conductance sits on exact multiples of \(e^2/h\). (b) Protection holds only while the system stays in the right phase, with a gap, at low enough temperature, and the required non-abelian anyons must first be shown to exist and be controllable, which is an experimental question still being argued.

What you learned¶

Smooth structures are finer than topology: there are 28 oriented smooth 7-spheres, and uncountably many smooth structures on \(\mathbb{R}^4\), while \(\mathbb{R}^n\) for \(n \ne 4\) has only one.
The smooth 4-dimensional Poincaré conjecture is open; four dimensions resist both the high-dimensional methods and the 3-dimensional ones.
Homotopy groups of spheres are simple to define and very hard to compute; the Hopf map shows \(\pi_3(S^2) \cong \mathbb{Z}\), and the stable groups begin \(\mathbb{Z}, \mathbb{Z}_2, \mathbb{Z}_2, \mathbb{Z}_{24}, 0, 0, \mathbb{Z}_2, \mathbb{Z}_{240}\).
In physics, quantised quantities such as the Hall conductance are topological invariants (Chern numbers); this was the subject of the 2016 Nobel Prize in Physics.
Topological quantum computing would compute with braids of anyons; whether it can be built is an open experimental question.
Many natural questions remain open, and some long-open ones have fallen recently.

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