CW Complexes and Cellular Approximation

7A.5 Computing π₁ computed fundamental groups by gluing: attaching a 2-dimensional disc along a loop kills that loop, and attaching higher-dimensional discs changes nothing. Spaces built entirely by attaching discs, dimension by dimension, are CW complexes. They are the natural home of algebraic topology: every smooth manifold has a CW structure (by Morse theory, 7A.13 Morse Theory), every simplicial complex is one, and almost every space in this guide is one up to homotopy. This chapter develops them and proves three facts that the rest of the book uses repeatedly.

• The fundamental group is read off from cells: generators from the 1-cells, relations from the attaching maps of the 2-cells. Higher cells do not matter.
• Cellular approximation: every map between CW complexes is homotopic to one that sends the $k$-skeleton into the $k$-skeleton. As a consequence, a sphere cannot be mapped essentially into a sphere of higher dimension: $\pi_k(S^n) = 0$ for $k < n$.
• Every finitely presented group is the fundamental group of a closed 4-manifold. So in dimension four, and above, the fundamental group can be anything a computer can write down, including groups whose triviality no algorithm can decide. Dimension three is different, and that difference is part of why the Poincaré conjecture is a statement about dimension three.

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

• build CW complexes by attaching cells, and give CW structures for spheres, projective spaces, surfaces and lens spaces;
• prove the homotopy extension property for CW pairs, and use it to collapse contractible subcomplexes, in particular maximal trees in graphs;
• compute $\pi_1$ of a CW complex from its 2-skeleton, and build a 2-complex with any prescribed fundamental group;
• state and prove cellular approximation (using Sard's theorem), and deduce $\pi_k(S^n) = 0$ for $k < n$ and that $\pi_1(X) = \pi_1(X^2)$;
• construct a closed 4-manifold with any finitely presented fundamental group, and explain why recognising the 3-sphere is algorithmically possible while recognising simply connected 4-manifolds is not;
• state Whitehead's theorem and use it to see what the Poincaré conjecture asks of homotopy theory.

6.1Recognising a sphere

6.2CW complexes

Write $D^n$ for the closed unit ball in $\mathbb{R}^n$, $S^{n-1} = \partial D^n$, and $e^n$ for the open ball, an open $n$-cell. Attaching an $n$-cell to a space $X$ along a map $\varphi : S^{n-1} \to X$ means forming the quotient $X\cup_\varphi D^n = (X\sqcup D^n)/(x \sim \varphi(x)\text{ for } x \in S^{n-1})$ (7A.1 Topological Spaces and Quotients).

The letters stand for "closure-finite" (each closed cell meets finitely many cells) and "weak topology", the two conditions in J. H. C. Whitehead's original 1949 definition, which the inductive construction guarantees. For finite complexes, which are compact, the weak topology is automatic. A subset $A \subseteq X$ that is a union of closed cells and itself a CW complex is a subcomplex, and $(X, A)$ a CW pair.

Two facts about the topology, which we state without proof: a compact subset of a CW complex meets only finitely many open cells; and CW complexes are Hausdorff, and locally contractible, so path-connected components and connected components agree and covering space theory (7A.7 Covering Spaces) applies.

The Euler characteristic of a finite CW complex is $\chi(X) = \sum_n(-1)^nc_n$, with $c_n$ the number of $n$-cells. For the sphere $1 + (-1)^n$, for $\mathbb{R}P^n$ it is $1$ or $0$, for the closed orientable surface of genus $g$ it is $2 - 2g$, matching 7A.3 Manifolds and Surfaces. That $\chi$ depends only on the space, not on the cell structure, is a theorem of homology (7A.10 Singular Homology and Mayer–Vietoris).

6.3Homotopy extension and collapsing

Graphs. A connected graph $\Gamma$ contains a maximal tree $T$, a contractible subgraph containing every vertex. Collapsing $T$ leaves one vertex with a loop for each edge not in $T$: $\Gamma \simeq \bigvee^kS^1$ with $k = E - V + 1 = 1 - \chi(\Gamma)$. By 7A.5 Computing π₁, $\pi_1(\Gamma)$ is free of rank $1 - \chi(\Gamma)$. Combined with covering spaces (7A.7 Covering Spaces), this proves the Nielsen–Schreier theorem: every subgroup of a free group is free, because it is $\pi_1$ of a covering space of a graph, which is a graph; and a subgroup of index $d$ in a free group of rank $k$ has rank $1 + d(k - 1)$, since a $d$-sheeted covering multiplies $\chi$ by $d$ (Exercise 6.14).

6.4The fundamental group of a CW complex

Examples: one 2-cell attached to a circle by a degree-$n$ map gives $\pi_1 = \mathbb{Z}/n$ (for $n = 2$ this is $\mathbb{R}P^2$); the torus is the presentation complex of $\langle a, b \mid aba^{-1}b^{-1}\rangle$. The dunce hat, a triangle with its three edges identified by the word $aaa^{-1}$, has $\pi_1 = \langle a \mid a\rangle = 1$, and it is in fact contractible, though it has no free face to start a simplicial collapse from: a standard example in combinatorial topology.

6.5Cellular approximation

A map between CW complexes is cellular if it sends $X^n$ into $Y^n$ for every $n$.

The heart of the proof is a single step.

Proof of the theorem (outline). Induct on skeleta of $X$. The 0-cells map into $Y$; move each to a 0-cell along a path (each point of $Y$ lies in a closed cell whose characteristic image is path-connected and meets $Y^0$). Having made $f$ cellular on $X^{k-1}$, consider a $k$-cell of $X$. Its image is compact, so meets finitely many cells of $Y$. Using the lemma, push it off each cell of dimension greater than $k$ that it meets, starting from the highest, keeping the boundary, which already lies in $Y^{k-1}$, fixed. Homotopy extension (Proposition 6.3) extends the homotopy from the $k$-skeleton to all of $X$. For infinite-dimensional $X$, perform the $k$-th stage during the time interval $[\frac{1}{2^{k+1}}, \frac{1}{2^k}]$.

This generalises 7A.4 The Fundamental Group, where $\pi_1(S^n) = 0$ for $n \geq 2$ was proved by a direct argument, and it is the input for the homotopy groups of 10A.4 Homotopy Groups, Degree and the Hurewicz Theorem. More generally, for any CW complex $X$, the inclusion $X^n \to X$ induces isomorphisms on $\pi_k$ for $k < n$ and a surjection for $k = n$: a sphere $S^k$ in $X$ can be made cellular, landing in $X^k$, and a homotopy $S^k\times I \to X$ can be made cellular, landing in $X^{k+1}$ (Exercise 6.15). The case $n = 2$, $k = 1$ is Theorem 6.5 again.

6.6Every finitely presented group in dimension four

Combined with a theorem of logic, this has striking consequences. Pyotr Novikov (1955) and William Boone (1958) proved that there are finitely presented groups with unsolvable word problem, and Sergei Adian and Michael Rabin (1955–58) deduced that no algorithm decides, given a finite presentation, whether the group it presents is trivial. Since the construction above is algorithmic, no algorithm decides whether a closed 4-manifold is simply connected, and Markov's theorem that homeomorphism of 4-manifolds is undecidable follows by refining the argument.

Dimension three is different. Fundamental groups of closed 3-manifolds are very special. Not every finitely presented group occurs: $\mathbb{Z}^4$ does not, for instance, and Peter Scott proved in 1973 that 3-manifold groups are coherent (finitely generated subgroups are finitely presented), a property that most finitely presented groups lack. After Perelman, geometrization implies that 3-manifold groups are residually finite and have solvable word problem. The Poincaré conjecture is the statement that the trivial group occurs in exactly one way: the only closed 3-manifold with trivial fundamental group is $S^3$. In dimension four the analogous statement, whether smooth homotopy 4-spheres are standard, is open (7A.12 The Poincaré Conjecture, Precisely).

6.7Whitehead's theorem and the shape of the conjecture

A map $f : X \to Y$ is a weak homotopy equivalence if it induces isomorphisms $\pi_k(X, x) \to \pi_k(Y, f(x))$ for all $k \geq 0$ and all base points (10A.4 Homotopy Groups, Degree and the Hurewicz Theorem defines $\pi_k$).

So for CW complexes, homotopy groups detect homotopy type. The proof uses cellular approximation and homotopy extension to build a homotopy inverse cell by cell (Hatcher, Theorem 4.5). It gives the homotopy-theoretic half of the Poincaré conjecture cheaply: a closed simply connected 3-manifold $M$ has homology $H_*(M) \cong H_*(S^3)$ by Poincaré duality (7A.11 Cohomology and Poincaré Duality), so by the Hurewicz theorem (10A.4 Homotopy Groups, Degree and the Hurewicz Theorem) $\pi_2(M) = 0$ and $\pi_3(M) \cong \mathbb{Z}$; a map $S^3 \to M$ representing a generator induces isomorphisms on all homology, hence (by a homology version of Whitehead's theorem) is a homotopy equivalence. Every closed simply connected 3-manifold is a homotopy 3-sphere. The whole difficulty of the Poincaré conjecture is to upgrade "homotopy equivalent to $S^3$" to "homeomorphic to $S^3$": a statement that no amount of homotopy theory can prove, since in other dimensions and categories it can fail (Milnor's exotic 7-spheres are homeomorphic but not diffeomorphic to $S^7$, 7A.12 The Poincaré Conjecture, Precisely).

6.8History

J. H. C. Whitehead introduced CW complexes in "Combinatorial homotopy I" (1949), together with his theorem on weak homotopy equivalences; cellular approximation is in the same work. Cell structures on projective spaces and lens spaces go back to the combinatorial topology of the 1920s and 1930s (Reidemeister, Seifert and Threlfall). The Nielsen–Schreier theorem dates from 1921–27. The unsolvability of the word problem is due to Novikov (1955) and Boone (1958), the triviality problem to Adian (1955) and Rabin (1958), and Markov's theorem on 4-manifolds to 1958. Scott's coherence theorem appeared in 1973, Rubinstein's 3-sphere recognition algorithm in 1992 and Thompson's proof in 1994.

6.9Exercises

Solution

For $(x, t)$ in the cylinder, follow the ray from $(0, 2)$ through $(x, t)$ until it first meets the bottom $D^n\times\{0\}$ or the side $S^{n-1}\times[0, 1]$. The ray is $(0, 2) + s((x, t) - (0, 2)) = (sx, 2 + s(t - 2))$, $s \geq 1$. It meets the bottom at $s_0 = \frac{2}{2 - t}$ if $|s_0x| \leq 1$, i.e. if $|x| \leq 1 - \frac t2$; otherwise it meets the side at $s_1 = \frac{1}{|x|}$. So $r(x, t) = \big(\frac{2x}{2 - t}, 0\big)$ if $|x| \leq 1 - \frac t2$, and $r(x, t) = \big(\frac{x}{|x|}, 2 - \frac{2 - t}{|x|}\big)$ otherwise. The two formulas agree on $|x| = 1 - \frac t2$, so $r$ is continuous, and it fixes the bottom and side. For $n = 1$, $x \in [-1, 1]$ and $\frac{x}{|x|} = \pm1$.

Solution

(a) $(z, 0) \mapsto (\zeta z, 0)$: the circle $|z| = 1$, $w = 0$ is invariant, and the action rotates it by $\frac{2\pi}{p}$, so the quotient is a circle $X^1$ (with one 0-cell and one 1-cell). (b) Hatcher's Example 2.43 gives a $\mathbb{Z}_p$-invariant cell structure on $S^3$ with $p$ cells in each dimension $0, 1, 2, 3$, permuted freely by the action, so the quotient has one cell in each dimension. The circle $\{w = 0\}$ is the union of the $p$ 0-cells and $p$ 1-cells, and each 2-cell is a disc whose boundary is that whole circle; in the quotient, the circle covers $X^1$ $p$ times, so the 2-cell is attached by a map of degree $p$. (c) One generator (the 1-cell), one relation $a^p$: $\mathbb{Z}/p$.

Solution

(a) Each vertex and each edge has exactly $d$ preimages, so $V$ and $E$ multiply by $d$. (b) $F_k = \pi_1$ of a wedge of $k$ circles, $\chi = 1 - k$. The subgroup is $\pi_1$ of the $d$-sheeted cover, a connected graph with $\chi = d(1 - k)$, so its $\pi_1$ is free of rank $1 - d(1 - k) = 1 + d(k - 1)$. (c) $d = 2$, $k = 2$: rank $3$. The cover has two vertices and four edges, each generator lifting to two edges going from one vertex to the other.

Solution

(a) Make the map cellular for the structure $e^0\cup e^k$ on $S^k$. (b) The homotopy $S^k\times I \to X$ is cellular on the subcomplex $S^k\times\{0, 1\}$; make it cellular rel that subcomplex; $S^k\times I$ has dimension $k + 1 \leq n$, so the homotopy lands in $X^{k+1} \subseteq X^n$. (c) Surjective by (a) (as $X^k \subseteq X^n$), injective by (b). (d) For $k = n$ the homotopy lands in $X^{n+1}$, where new $(n+1)$-cells can kill classes: $\pi_1(S^1) = \mathbb{Z}$ but in $D^2 = S^1\cup e^2$ the circle becomes trivial.

Solution

(a) A generic map of a $k$-dimensional complex into a 4-manifold misses a given 1-dimensional submanifold when $k + 1 < 4$, so for $k = 1$ (loops) and $k = 2$ (homotopies): $\pi_1(M\setminus\gamma) \to \pi_1(M)$ is onto and injective. (b) Using the cell structures $D^2 = S^1\cup e^2$ and $S^2 = e^0\cup e^2$, $D^2\times S^2$ is obtained from its boundary $S^1\times S^2$ by attaching the product cells $e^2\times e^0$, a 2-cell attached along $S^1\times\{\mathrm{pt}\}$ (which is $\gamma$ pushed to the boundary), and $e^2\times e^2$, a 4-cell. The 2-cell kills $\gamma$, the 4-cell changes nothing (7A.5 Computing π₁).

Solution

(a) All three vertices are identified (follow the edge identifications), so one 0-cell; all edges are the same 1-cell $a$; the triangle is the 2-cell, attached along $aaa^{-1}$. $\pi_1 = \langle a \mid aaa^{-1}\rangle = \langle a \mid a\rangle = 1$. (b) $1 - 1 + 1 = 1$. (c) $D$ is simply connected with $H_2 = 0$ and no higher cells, so by Hurewicz all homotopy groups vanish, and the map to a point is a weak equivalence, hence a homotopy equivalence by Whitehead.