Aspherical manifolds

In this post I want to sketch the idea of aspherical manifolds - manifolds which don't admit higher homotopically non-trivial spheres - and the related concepts of Eilenberg-MacLane-spaces and classifying spaces for groups.

DefinitionA topological space $M$ is called aspherical if all higher homotopy groups vanish, i.e. $\pi_n(M,m_0) = 0 \quad \forall n > 1$ where $m_0 \in M$ is an arbitrary basepoint and $M$ is assumed to be connected.

Since manifolds admit universal covers, you could equivalently define a manifold to be aspherical if and only if its universal cover is contractible.

Just one example illustrating how rich this class of spaces is:
Metric spaces that are of non-positive curvature (i.e. locally CAT(0)-spaces), for example the Bruhat-Tits building of a simple algebraic group over a field with a discrete valuation, are aspherical.

A good survey on aspherical manifolds was given by Wolfgang Lück.

DefinitionA connected topological space $X$ is called Eilenberg-MacLane-space for a group $G$ and a natural number n if its nth homotopy group is exactly $G$ and all other homotopy groups vanish, i.e.
$\pi_k(X,x_0) = \begin{cases} G & k=n \\ 0 & else.\end{cases}$
Then one calls $X$ also $K(G,n)$.

The standard examples of $K(G,1)$ spaces are $S^1$, which is a $K(\mathbb{Z},1)$ and $\mathbb{R}P^\infty$, which is a $K(\mathbb{Z}/2,1)$.
Of course, every $K(G,1)$ is aspherical and every aspherical space is a $K(G,1)$ for $G$ being its fundamental group.

One can also define a functorial construction of a $K(G,1)$ which gives a CW-complex model for every group $G$ and transforms group homomorphisms into continuous maps of spaces.

For this, we need the functorial nerve construction.
DefinitionThe nerve $N(G)$ of a (discrete) group $G$ is the simplicial $G$-set with n-simplices being the (n+1)-fold cartesian product of sets $G \times G \times \cdots \times G$, face maps just omitting one factor in the cartesian product, degeneracies adding the identity element of $G$ in one factor.
By construction, seen as a discrete simplicial group, $G$ embeds into $N(G)$ as the 0-skeleton. Observe that $N(G)$ is contractible, since every n-simplex $(g_0,...,g_n) \in N(G)_n$ is the face of $(e,g_0,...,g_n) \in N(G)_{n+1}$ which also has the face $(e,g_1,...,g_n) \in N(G)_n$, thus allowing to move every point to the identity $(e,...,e) \in N(G)_m$ which is just a degeneracy of $e \in N(G)_0 = G$.
The group $G$ acts diagonally on $N(G)$, i.e. it acts on an n-simplex by the formula $(g,(g_0,...,g_n)) \mapsto (gg_0,...,gg_n) \in N(G)_n$. This action is compatible with face and degeneracy maps, thus making $N(G)$ into a simplicial $G$-set. The action is free, i.e. no two elements of $G$ operate in the same way.

Using the nerve construction, we now define the classifying space:
DefinitionThe classifying space $BG$ of a group $G$ is the quotient $BG := |N(G)|/G$ of the geometric realisation $|N(G)$ of the nerve construction by the group action described above. It turns out that $G$ operates on $|N(G)|$ like a deck transformation group, thus giving $BG$ the structure of a CW-complex with universal cover $|N(G)|$ and fundamental group $G$.
A group homomorphism $\phi : G \to H$ gives rise to a morphism of simplicial sets $N(\phi) : N(G) \to N(H)$ by pointwise application. Geometric realisation is also functorial, and due to $\phi$ being a homomorphism, the continuous map $|N(\phi)| : |N(G)| \to |N(H)|$ descends to a continuous map of classifying spaces $B\phi : BG \to BH$.

If you are not into simplicial sets and geometric realisation, you can look for a more hands-on approach in Hatcher's book "Algebraic Topology", on page 87, chapter 1.B, more specifically Example 1B.7 on page 89.

Now back to our first definitions: An aspherical manifold is just a manifold which happens to be a $K(G,1)$ for $G$ being its fundamental group. The classifying space is just an explicit (functorial!) construction which gives a $K(G,1)$ for every group $G$ (although most authors would call our $BG$ just one explicit model for $BG$...).

One would like to work only with CW-complexes, if possible, since they allow induction over the skeleton and cell-by-cell arguments. Is every manifold homeomorphic to a CW-complex - long time ago there was the "Hauptvermutung" (main conjecture) which asked this, but it's wrong. While compact manifolds admit a homotopy equivalent CW-model (by Kirby and Siebenmann), this is not true for topological manifolds in general. Let us look what one could do with a CW-model:

PropositionLet $X$ be a connected CW complex and $Y$ be a $K(G,1)$ (for example, your favourite aspherical manifold). Let $\phi : \pi_1(X,x_0) \to \pi_1(Y,y_0) = G$ be a homomorphism of groups. Then there is a continuous map $\Phi : X \to Y$ mapping $x_0$ to $y_0$ which induces $\phi$ on fundamental groups; furthermore, the map $\Phi$ is unique up to homotopy relative $x_0$.

The proof of this proposition goes roughly like that: First, let $\Phi$ map $x_0$ to $y_0$. Now, for each 1-cell $\gamma$, take a representative of $\phi([\overline{\gamma}]) \in \pi_1(Y,y_0)$ to define $\Phi$ on $\gamma$. Then one has to extend the map given on the 1-skeleton to $X$, using the fact that $Y$ has no higher homotopy.

CorollaryEvery two CW-complexes $X,Y$ which are both $K(G,1)$-spaces are homotopy equivalent ("of the same homotopy type").

To prove this, just take isomorphisms $f : \pi_1(X,x_0) \to G$ and $g : \pi_1(Y,y_0) \to G$ and define $\phi := f \circ g^{-1}$ which gives $\Phi : Y \to X$ with inverse up to homotopy given by $\Psi : X \to Y$ induced by $\psi := g \circ f^{-1}$.

This justifies that every invariant of $BG$ that depends only on the homotopy type, is actually an invariant of $G$ - a very useful idea. One can define group homology with integer coefficients of $G$ by the formula $H_n(G,\mathbb{Z}) := H_n(BG,\mathbb{Z})$.

One drawback of the classifying space via the nerve construction is that it is usually very large - there are simplices in arbitrary high dimensions. For example, the circle $S^1$, given as example of a $K(\mathbb{Z},1)$, is much more efficient than $B\mathbb{Z}$.

Of course, talking about aspherical manifolds, we don't want to forget the manifold structure. Given a group $G$, one could expect that many non-homeomorphic aspherical manifolds with fundamental group $G$ exist - even many non-homotopy equivalent ones. At least we can say that such non-homotopy equivalent aspherical manifolds are not of CW homotopy type. There is an old conjecture on this theme:

Conjecture (Borel)Let M and N be closed aspherical manifolds, and let $f : M \to N$ be a homotopy equivalence. Then $f$ is homotopic to a homeomorphism.

Together with the result of Kirby and Siebenmann (that every closed manifold is of CW homotopy type), this would imply that closed aspherical manifolds are classified by their fundamental group up to homeomorphism.

The property that every homotopy equivalence is homotopic to a homeomorphism is called topological rigidity.