Homotopy Groups, Degree and the Hurewicz Theorem

The fundamental group detects loops that cannot be contracted. The Poincaré conjecture is a statement about $\pi_1$, but its proof by the Ricci flow runs on spheres: 2-spheres that cannot be shrunk, and families of 2-spheres sweeping out a 3-manifold that cannot be pulled tight. Perelman's third preprint, and Morgan and Tian's chapter 18 which expands it, use $\pi_2(M)$ and $\pi_3(M)$ at every step. First they show that the flow with surgery kills $\pi_2$: after a finite time every component has $\pi_2 = 0$. Then they use a nonzero element of $\pi_3$, represented as a 2-sphere's worth of loops, to define a min–max quantity that must become negative in finite time, so that the flow must become extinct.

This chapter builds exactly that algebraic topology, from the definitions. It proves what is short enough to prove: the higher homotopy groups are abelian; they are unchanged by coverings; maps from lower-dimensional spheres to higher-dimensional ones are null-homotopic; and degree classifies maps between spheres of equal dimension up to the point where it matters. It states, carefully, the deeper facts that are used: the long exact sequence of a fibration, the Hurewicz theorem and Whitehead's theorem. And it computes the groups that the finite extinction argument needs for every closed 3-manifold that arises.

By the end of this chapter you will be able to:

• define $\pi_n(X, x_0)$, prove that it is abelian for $n \geq 2$, and explain the action of $\pi_1$;
• prove that a covering induces isomorphisms on $\pi_n$ for $n \geq 2$, and compute $\pi_n$ of tori, lens spaces and $S^2\times S^1$;
• prove that $\pi_k(S^n) = 0$ for $k < n$ with Sard's theorem, and relate $\pi_n(S^n)$ to degree;
• compute $\pi_2(S^2)$ and $\pi_3(S^2)$ from the Hopf fibration;
• state the Hurewicz and Whitehead theorems and use them to compute $\pi_2$ and $\pi_3$ of closed 3-manifolds, and to recognise aspherical ones;
• explain how an element of $\pi_3(M)$ becomes a sweepout by 2-spheres, or a 2-sphere's worth of loops, as in Colding–Minicozzi's and Perelman's arguments.

4.1Higher homotopy groups

Let $X$ be a space with a base point $x_0$, and $n \geq 1$. Write $I^n = [0, 1]^n$ and $\partial I^n$ for its boundary.

Collapsing $\partial I^n$ to a point turns $I^n$ into the sphere $S^n$, so $\pi_n(X, x_0)$ is equally the set of based homotopy classes of maps $(S^n, \ast) \to (X, x_0)$. For $n = 1$ this is the fundamental group of 7A.4 The Fundamental Group, and the proofs that $+$ is well defined, associative, with identity the constant map and inverse $f(1 - s_1, s_2, \ldots)$, are the same as there: they only ever reparametrise the first coordinate. A map $X \to Y$ sending $x_0$ to $y_0$ induces homomorphisms $\pi_n(X, x_0) \to \pi_n(Y, y_0)$ by composition, and homotopic maps (through based maps) induce the same homomorphism. So homotopy equivalences induce isomorphisms.

For $n = 1$ there is no room to move the loops past each other, and indeed $\pi_1$ need not be abelian. This picture also explains a fact used constantly below. A free homotopy of maps $S^n \to X$ may move the base point, and following the base point along a loop $\gamma$ changes a based class $[f]$ to a class $\gamma\cdot[f]$: $\pi_1$ acts on $\pi_n$. Free homotopy classes of maps $S^n \to X$ correspond to orbits of this action. When $X$ is simply connected the action is trivial, and free and based classes are the same thing. Perelman's and Morgan–Tian's "nontrivial element of $\pi_2(M)$", represented by any map $S^2 \to M$ in the free homotopy class, uses this.

4.2Coverings do not see higher homotopy

In particular, $\pi_n(X) \cong \pi_n(\tilde X)$ for the universal cover, for all $n \geq 2$: the higher homotopy groups of $X$ are those of a simply connected space. Examples:

• $\pi_n(S^1) = \pi_n(\mathbb{R}) = 0$ for $n \geq 2$, and $\pi_n(T^3) = \pi_n(\mathbb{R}^3) = 0$ for $n \geq 2$. More generally every closed hyperbolic or flat manifold, being covered by a contractible space (Cartan–Hadamard, 9A.7 Jacobi Fields and Curvature versus Topology), has $\pi_n = 0$ for all $n \geq 2$.
• $\pi_n(S^3/\Gamma) = \pi_n(S^3)$ for every spherical space form (10A.2 Finite Rotation Groups and Spherical Space Forms) and $n \geq 2$.
• $S^2\times S^1$ is covered by $S^2\times\mathbb{R}$, which deformation retracts onto $S^2$, so $\pi_n(S^2\times S^1) = \pi_n(S^2)$ for $n \geq 2$.
• $\mathbb{RP}^3\#\mathbb{RP}^3$ is double covered by $S^2\times S^1$ (Exercise 4.13), so its universal cover is $S^2\times\mathbb{R}$ and $\pi_2(\mathbb{RP}^3\#\mathbb{RP}^3) = \pi_2(S^2) = \mathbb{Z}$.

So far we have reduced everything to simply connected spaces. To compute, we need the groups of spheres.

4.3Maps between spheres

The proof is the smooth version of a general principle, cellular approximation: a map from a $k$-dimensional complex into a space built from cells can be deformed into the cells of dimension at most $k$. For our purposes the smooth version is enough.

Maps $S^n \to S^n$ of the same dimension are measured by their degree (7A.8 Smooth Topology): for a smooth map, the number of preimages of a regular value counted with signs $\pm1$ according to whether the differential preserves or reverses orientation. Degree is independent of the regular value and invariant under homotopy. Equivalently, $f_*$ acts on $H_n(S^n) \cong \mathbb{Z}$ as multiplication by $\deg f$. The same definitions work for maps between any two closed, connected, oriented $n$-manifolds $M \to N$, and $f_*[M] = (\deg f)[N]$ on fundamental classes.

That every integer occurs is easy: on $S^1$, $z \mapsto z^k$ has degree $k$, and suspending (extending a map of the equator $S^{n-1}$ to $S^n$ by coning towards the two poles) preserves degree, so every degree occurs on every $S^n$. The difficult half is that degree is a complete invariant; Milnor's book proves it with framed cobordism.

For the Ricci flow, degree enters through degree-one maps. A closed oriented 3-manifold $M$ with finite fundamental group has universal cover $\tilde M$ a closed simply connected 3-manifold, and the covering $\tilde M \to M$ has degree $|\pi_1(M)|$. If $M$ is simply connected, the next sections show that a generator of $\pi_3(M)$ is a map $S^3 \to M$ of degree $\pm1$, and that such a map is a homotopy equivalence: a simply connected closed 3-manifold is a homotopy 3-sphere. The Poincaré conjecture is the statement that it is in fact diffeomorphic to $S^3$.

4.4The Hopf fibration computes $\pi_3(S^2)$

Maps between spheres of different dimensions can be essential when the target has the lower dimension. The first and most important example is the Hopf map $h : S^3 \to S^2$ of 10A.1 A Zoo of Three-Manifolds. It is a fibre bundle: every point of $S^2$ has a neighbourhood $U$ with $h^{-1}(U) \cong U\times S^1$ compatibly with $h$. Fibre bundles, like coverings (whose fibres are discrete), have a homotopy lifting property, and that gives the following tool.

The map $\partial$ is defined by lifting: a map $(I^n, \partial I^n) \to (B, b_0)$ is lifted to $E$, starting at $e_0$ on one face, and its restriction to the opposite face lands in the fibre $F$, giving an element of $\pi_{n-1}(F)$. A covering is a fibre bundle with discrete fibre, and the sequence then reproduces Proposition 4.3.

For the Hopf fibration $S^1 \to S^3 \to S^2$:

• $\pi_2(S^3) = 0$ and $\pi_1(S^3) = 0$ give $0 \to \pi_2(S^2) \xrightarrow{\partial} \pi_1(S^1) \to 0$, so $\pi_2(S^2) \cong \pi_1(S^1) = \mathbb{Z}$, in agreement with the degree.
• $\pi_3(S^1) = 0$ and $\pi_2(S^1) = 0$ (the universal cover of $S^1$ is $\mathbb{R}$) give $0 \to \pi_3(S^3) \xrightarrow{h_*} \pi_3(S^2) \to 0$, so

generated by the Hopf map itself.

The Hopf map is therefore an essential map from a sphere to a lower-dimensional one, a phenomenon with no analogue for $\pi_1$. It is detected by the Hopf invariant: the preimages of two regular values are two circles in $S^3$, and their linking number is a homotopy invariant, equal to $1$ for $h$ (two Hopf fibres are linked once, 10A.1 A Zoo of Three-Manifolds). The consequence for 3-manifolds: $\pi_3(S^2\times S^1) \cong \pi_3(S^2) \cong \mathbb{Z}$, so even $S^2\times S^1$, which has infinite fundamental group, has nonzero $\pi_3$.

4.5The Hurewicz and Whitehead theorems

Homology is computable, homotopy groups are not, in general. The bridge is a natural homomorphism, the Hurewicz map $h_n : \pi_n(X) \to H_n(X)$, sending $[f : S^n \to X]$ to $f_*[S^n]$, the image of the fundamental class.

Two remarks about the hypotheses. Hurewicz needs the lower homotopy groups to vanish, not the lower homology groups; the converse direction holds only with $\pi_1 = 1$ added: a simply connected space whose homology vanishes below degree $n$ has $\pi_k = 0$ below $n$ (the "homology Hurewicz" form, proved by induction from the theorem). Whitehead needs a map inducing the isomorphisms: two spaces with isomorphic homotopy groups need not be homotopy equivalent.

4.6The homotopy groups of closed 3-manifolds

Now everything can be computed for the manifolds the flow meets. Let $M$ be a closed, connected, orientable 3-manifold.

The dichotomy in part 2 is sharp: a closed 3-manifold with $\pi_2 = 0$ has either finite fundamental group, and then $\pi_3 = \mathbb{Z}$, or infinite fundamental group, and then every higher homotopy group vanishes. There is nothing in between. This is why the finite extinction argument needs a hypothesis on $\pi_1$: on an aspherical manifold, such as a hyperbolic one or the 3-torus, there is no nonzero element of $\pi_3$ to define a min–max quantity with, and indeed the flow does not become extinct there (hyperbolic metrics expand forever, 10A.10 Geometrization and Ricci Flow).

An embedded sphere that is nonzero in $\pi_2$ does not bound a ball (a sphere bounding a ball is contractible through the ball). So, combined with 10A.5 The Prime Decomposition: an irreducible orientable 3-manifold has $\pi_2 = 0$, and a closed orientable 3-manifold with $\pi_2 \neq 0$ either has an $S^2\times S^1$ summand or is a nontrivial connected sum. Conversely, a nontrivial connected sum $P\#Q$ of two manifolds with nontrivial fundamental groups has $\pi_2 \neq 0$: in the universal cover the separating sphere lifts to spheres separating it into noncompact pieces. A sphere $\Sigma$ that separates a simply connected manifold into two noncompact pieces is not null-homologous: a proper line running from one noncompact piece to the other can be chosen to cross $\Sigma$ once, and the intersection number of a closed 2-cycle with a proper line is unchanged by adding boundaries, so it would be $0$ for a null-homologous $\Sigma$. So $\Sigma$ is nonzero in $H_2$ of the cover, which is $\pi_2$ by Hurewicz. ($\mathbb{RP}^3\#\mathbb{RP}^3$ above is an example.)

4.7Spheres and loops in the extinction argument

Morgan and Tian's Theorem 18.1 assumes that $\pi_1(M)$ is a free product of finite groups and infinite cyclic groups; simply connected manifolds are the main case. The argument has two stages, and this chapter supplies the topology of each.

Stage 1: killing $\pi_2$. For a metric $g$ and a nonzero class $\beta \in \pi_2(M)$, let $W_2(\beta, g)$ be the infimum of the energies $\frac12\int|d\sigma|^2$ (or areas) of maps $\sigma : S^2 \to M$ in the free homotopy class $\beta$. By Sacks and Uhlenbeck the infimum is attained by a minimal (harmonic) sphere, or a union of them, and the evolution of area of 10A.11 Min–Max and Width gives a differential inequality that forces $W_2$ to become negative in finite time unless the class dies. Surgery does not increase the quantity. So after finite time every component has $\pi_2 = 0$.

Stage 2: using $\pi_3$. Now every component $N$ has $\pi_2(N) = 0$. Surgery only splits off connected summands (10A.5 The Prime Decomposition), so the fundamental group of each component is, like that of $M$, a free product of finite and infinite cyclic groups. A compact 3-manifold whose fundamental group is a nontrivial free product, or infinite cyclic, contains a homotopically nontrivial embedded 2-sphere (this combines the sphere theorem with Kneser's conjecture, proved by Stallings; statement). As $\pi_2(N) = 0$, neither can happen, so $\pi_1(N)$ is a single finite group, and by Proposition 4.9 $\pi_3(N) \cong \mathbb{Z} \neq 0$. This is the argument Morgan and Tian give at the start of their chapter 18. A nonzero element of $\pi_3$ can be presented in two ways, and the two approaches to finite extinction use one each.

• As a path of 2-spheres (Colding–Minicozzi). Write $S^3$ as $S^2\times[0, 1]$ with $S^2\times\{0\}$ and $S^2\times\{1\}$ collapsed to points (the suspension of $S^2$). A map $S^3 \to N$ is then a path $s \mapsto \sigma_s$ of maps $S^2 \to N$ starting and ending at constant maps: a sweepout. Nontrivial elements of $\pi_3$ are nontrivial sweepouts, and the width of 10A.11 Min–Max and Width is the min–max of energy over them.
• As a 2-sphere of loops (Perelman). Write $S^3$ instead as the join, or think of a map $S^2\times S^1 \to N$ that collapses $S^2\times\{\ast\}$ and $\{\ast\}\times S^1$ to the base point: it is the same as a map $S^2 \to \Omega N$, a 2-sphere's worth of based loops. This adjunction gives $\pi_3(N) \cong \pi_2(\Omega N)$. Perelman uses the free loop space $\Lambda N$ of all loops, which fibres over $N$ (evaluate at the base point of the loop) with fibre $\Omega N$ and has a section (constant loops). The long exact sequence then splits:

and with $\pi_2(N) = 0$ this is $\pi_3(N)$. Each loop in a family representing a nonzero element of $\pi_2(\Lambda N)$ bounds discs (it is null-homotopic, since by now we may pass to the simply connected universal cover or note the loops are contractible), and Perelman's invariant is the infimum over families of the maximal area of least-area discs spanning the loops (10A.11 Min–Max and Width, 12C.3 Finite Extinction).

In both presentations, nontriviality is what prevents the families from being pulled tight to constants, and the min–max value is positive for that reason; the Ricci flow then forces it to decrease at a definite rate, which is impossible for all time.

4.8History

Witold Hurewicz introduced the higher homotopy groups in 1935 and proved the theorem named after him; Eduard Čech had defined them earlier, in 1932, but abandoned them when they turned out to be abelian. Heinz Hopf discovered the essential map $S^3 \to S^2$ in 1931 and proved the degree theorem in the 1920s. J. H. C. Whitehead proved his theorem in 1949. Christos Papakyriakopoulos proved the sphere theorem, with Dehn's lemma and the loop theorem, in 1957; J. H. C. Whitehead simplified part of the argument. The use of $\pi_3$ in finite extinction is from Perelman's third preprint (2003) and Colding and Minicozzi (2005).

4.9Exercises

Solution

A map $S^n \to X\times Y$ is a pair of maps $S^n \to X$, $S^n \to Y$, and a homotopy of such maps is a pair of homotopies, so based homotopy classes of maps into the product are pairs of classes; the product structure is componentwise. Then $\pi_n(S^2\times S^1) = \pi_n(S^2)\times\pi_n(S^1) = \pi_n(S^2)$ for $n \geq 2$, since $\pi_n(S^1) = 0$; so $\pi_2 = \mathbb{Z}$ (degree) and $\pi_3 = \mathbb{Z}$ (Hopf).

Solution

$\pi_2(T^3) = \pi_3(T^3) = 0$, since the universal cover $\mathbb{R}^3$ is contractible (Proposition 4.3); so every based map is null-homotopic, and a free map is homotopic to a based one by moving its base point along a path. A null-homotopic map has the degree of a constant map, $0$: the induced map on $H_3$ factors through $H_3(\text{point}) = 0$. The collapse map $c : T^3 \to T^3/(T^3\setminus B) \cong S^3$, for a small ball $B$, is a homeomorphism from $B$ onto its image away from one point, so a regular value inside the image of $B$ has exactly one preimage, at which $c$ preserves orientation: $\deg c = 1$. So degree-one maps $M \to S^3$ exist for every $M$, while degree-one maps $S^3 \to M$ exist only for special $M$.

Solution

$\tau(x, z) = (x, z)$ would need $-x = x$, impossible, so the action is free. The circle maps $z \mapsto \bar z$ fixing $z = \pm1$; the arc $A = \{\operatorname{Im}z \geq 0\}$ from $1$ to $-1$ is a fundamental domain for $z \mapsto \bar z$. So the quotient is $S^2\times A$ with identifications only over the endpoints: over $z = 1$, the sphere $S^2\times\{1\}$ is divided by the antipodal map, and similarly over $z = -1$. A neighbourhood of $S^2\times\{1\}$ in the quotient is $(S^2\times[0, \delta))/\!\sim$ with the antipodal identification at $0$: this is the mapping cylinder of $S^2 \to \mathbb{RP}^2$, which is $\mathbb{RP}^3$ minus a ball. So the quotient is two copies of $\mathbb{RP}^3$ minus a ball, joined along $S^2\times(\text{middle of }A)$: $\mathbb{RP}^3\#\mathbb{RP}^3$. Its fundamental group is $\mathbb{Z}/2 * \mathbb{Z}/2$, infinite, and its universal cover is that of $S^2\times S^1$, namely $S^2\times\mathbb{R}$.

Solution

$\pi_3(L(p, q)) \cong \pi_3(S^3) = \mathbb{Z}$, generated by the covering map $c : S^3 \to L(p, q)$ (the lift of $c$ to the universal cover is the identity of $S^3$, a generator). The covering has $p$ sheets and preserves orientation, so a regular value has $p$ preimages, all positive: $\deg c = p$, that is $c_*[S^3] = p[L(p, q)]$. So the Hurewicz map sends the generator to $p$ times the generator of $H_3 = \mathbb{Z}$. The theorem requires $\pi_1 = \pi_2 = 0$ to conclude that $h_3$ is an isomorphism; here $\pi_1 = \mathbb{Z}/p \neq 0$.

Solution

Let $H$ be a free homotopy from $f$ to $g$ and $\gamma(t) = H(\ast, t)$, a loop at $x_0$. Since $X$ is simply connected, $\gamma$ is null-homotopic rel endpoints by some $G : [0, 1]\times[0, 1] \to X$ from $\gamma$ to the constant loop. The homotopy extension property of the pair $(S^n\times[0, 1], \{\ast\}\times[0, 1])$, which holds because $\{\ast\}$ is a subcomplex, lets one deform $H$ so that its restriction to $\{\ast\}\times[0, 1]$ follows $G$ and ends constant, keeping the ends $f$ and $g$ fixed. The result is a based homotopy. Without simple connectivity the loop $\gamma$ cannot be contracted, and $g$ is homotopic only to $\gamma\cdot f$: this is the action of $\pi_1$ on $\pi_n$.

Solution

(a) At $s = 0$ and $s = \pi$ every $x$ goes to $(\pm1, 0)$, the poles. For $0 < s < \pi$, $(\cos s, \sin s\,x)$ determines $s$ (from the first coordinate) and $x$ (from the rest, divided by $\sin s > 0$), so the map from the suspension of $S^2$ is a continuous bijection from a compact space to a Hausdorff one: a homeomorphism, of degree $\pm1$, hence nontrivial in $\pi_3(S^3)$. (b) $\sigma_s(S^2)$ is the sphere of latitude $s$, a round 2-sphere of radius $\sin s$, with area $4\pi\sin^2s \leq 4\pi$, equality at $s = \frac\pi2$, where it is the totally geodesic equator. (c) Lengths scale by $\sqrt{1 - 4t}$ and areas by $1 - 4t$: $A(t) = 4\pi(1 - 4t)$, so $A' = -16\pi$. The general inequality uses $R \geq R_{\min}$; here $R = \frac{6}{1 - 4t} > 0$, so the term $-\frac12R_{\min}A = -\frac12\cdot\frac{6}{1 - 4t}\cdot4\pi(1 - 4t) = -12\pi$ adds to the $-4\pi$ from Gauss–Bonnet: $A' = -4\pi - 12\pi = -16\pi$, with equality because the equator has $|A|^2 = 0$ and $\operatorname{Ric}(\nu, \nu)$ is constant. (The inequality of 10A.11 Min–Max and Width drops the helpful term $-\frac12R_{\min}A$ when $R_{\min} > 0$ and replaces it with the universal bound; it is far from sharp on the sphere.)