Essential manifolds

Now I'll explain a little bit what essential manifolds are and what they're good for.

DefinitionA (connected closed orientable topological) n-manifold $M$ is called essential, if there exists a continuous map $f : M \to K(\pi_1(M,\ast),1)$ such that the induced morphism on the top homology $f_\ast : H_n(M,\mathbb{Z}) \to H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$ maps the fundamental class $[M] \in H_n(M,\mathbb{Z})$ to some non-zero element $f_\ast([M]) \neq 0 \in H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$.

To have a very explicit example, take a n-torus $M$, that is a manifold of dimension n which is homotopy equivalent to a product of n copies of $S^1$. Each such $S^1$ yields a different non-contractible loop on $M$, so there are n non-homotopic loops $\gamma_1,...,\gamma_n$ and the fundamental group is just $\pi_1(M,\ast) = \mathbb{Z}[\gamma_1,...,\gamma_n]$, the free abelian group generated by the $\gamma_i$. The homology is the exterior algebra over the fundamental group. The cohomology is the exterior algebra over the dual of the fundamental group, i.e. $H^\bullet(M,\mathbb{Z}) = \mathbb{Z}[\gamma_1^\ast,...,\gamma_n^\ast]$. The fundamental class is just $\gamma_1 \wedge ... \wedge \gamma_n \in H_n(M,\mathbb{Z})$. The universal cover of a n-torus is n-dimensional euclidean space, which is contractible, so $M$ has a contractible universal cover, thus it is acyclic, in other words, a $K(\pi_1(M,\ast),1)$. Taking the identity map $f := id_M$, this induces on top homology the identity map (since homology is functorial) and thus maps the fundamental class to itself, a non-zero element. So we have seen that any torus is essential. Note that we haven't looked at metric properties at all, because essentialness is a purely homotopy theoretic notion.

If you look closer, you see that we haven't actually used that the space $M$ was a torus - we just used that it is an aspherical space, so every aspherical manifold is essential.

The Borel conjecture predicts that closed aspherical manifolds are topologically rigid. The most common examples of non-topologically rigid spaces are lens spaces - there are many non-homeomorphic lens spaces of the same homotopy type. Lens spaces are closed, and they are good examples of non-aspherical essential manifolds, so they don't disprove the Borel conjecture.

DefinitionLet $p$ and $q_1,...,q_n$ be integers (for some $n \geq 2$), with $q_i$ coprime to $p$ for each $i$. Define $\ell_k := 2\pi i q_k/p$. Take the unit sphere in $\mathbb{C}^n$, which is a $S^{2n-1}$ and let $\mathbb{Z}/p$ act on it by $$[1].(z_1,...,z_n) := (e^{\ell_1}z,...,e^{\ell_n}z).$$
The quotient of $S^{2n-1}$ by this action is denoted $L(p;q_1,...,q_n)$, the {lens space associated to $(p;q_1,...,q_n)$.

This is a $(2n-1)$-dimensional closed manifold with fundamental group $\mathbb{Z}/p$. The universal cover is given by the quotient map $S^{2n-1} \to L(p;q_1,...,q_n)$, so the universal cover is clearly non-contractible and in fact very spherical. This shows that lens spaces are never aspherical.

In the literature on homology and homotopy, you'll often find 3-dimensional lens spaces $L(p,q) := L(p;1,q)$. For these, there exists a nice classification of homeomorphism types via Reidemeister torsion (or: simple homotopy type), ultimately boiling down the question to arithmetic relation between different $q$, modulo $p$.

To see that lens spaces are essential, we have to produce a map $f : L(p;q_1,...,q_n) \to K(\mathbb{Z}/p,1)$ which on top homology maps the fundamental class to a non-zero element. The homology of $K(\mathbb{Z}/p,1)$ is well-known, it is
$$H_k(\mathbb{Z}/p,\mathbb{Z}) = \begin{cases} \mathbb{Z} & k=0,\\ \mathbb{Z}/p & k \text{ odd},\\ 0 & k \text{ even}. \end{cases}$$
The dimension of a lens space is $2n-1$, so it is odd - phew!

Now we need an explicit model for $K(\mathbb{Z}/p,1)$. One such model is given by the infinite lens space $L^\infty(p) := S^\infty/_{\mathbb{Z}/p}$, where $S^\infty := \lim S^n$ is seen as the union of spheres where the n-sphere sits inside the (n+1)-sphere as equator. The group $\mathbb{Z}/p$ acts by multiplication with p-th roots of unity in each coordinate, which is possible by putting the $S^\infty$ in a $\mathbb{C}^\infty := \lim \mathbb{C}^n$ by taking the limit over the embeddings $S^{2n-1} \to \mathbb{C}^n$.
We can modify this construction slightly, by starting with the lens space $L(p;q_1,...,q_n)$ and taking the limit over all $L(p;q_1,...,q_n,q'_1,...,q'_k)$ for $k \to \infty$ and $q'_i = q_n$ for all i. This yields the same $L^\infty(p)$ up to homotopy and even better, it admits an inclusion map from $L(p;q_1,...,q_n)$. On homology, the inclusion map maps the fundamental form to a generator of $\mathbb{Z}/p$, which is non-zero. Therefore, lens spaces are essential.

With a very similar idea, one can prove that real projective spaces $\mathbb{R}P^n$ are essential, by looking at the inclusion into $\mathbb{R}P^\infty = \lim \mathbb{R}P^k$, which is aspherical with the same fundamental group $\mathbb{Z}/2$.

In general, it suffices to find a continuous map of non-zero degree from a manifold $M$ onto an essential manifold to deduce that $M$ is essential.

To give a counter-example, look at the spherical space $S^n$ (for $n \geq 2$) with trivial fundamental group. It is certainly not aspherical (its higher homotopy groups are quite interesting) but there is an inclusion map $S^n \to S^\infty$ (as above). This inclusion map has to be the zero map on top degree homology, since $H_n(S^\infty,\mathbb{Z}) = 0$ for all $n \geq 1$ (because $S^\infty$ is contractible). This shows that spheres are never essential.

Finally, you might ask
What are essential manifolds good for?In his 1983 paper "Filling Riemannian Manifolds", Gromov defined essential manifolds the first time, to state (and prove) his "main isosystolic inequality".
To formulate it, we have to say what a systole is first:

DefinitionLet $M$ be a Riemannian manifold. Then the systole of $M$ is $sys_1(M) := \inf_{\gamma} length(\gamma)$, where the infimum goes over all non-contractible loops $\gamma$ in $M$ (in fact it is a minimum).

Theorem (Gromov)Let $M$ be a closed essential Riemannian manifold of dimension $n$. Then
$$sys_1(M) \leq C_n \sqrt[n]{Vol(M)}$$ with some constant $C_n$ not depending on $M$ which satisfies
$$0 < C_n < 6(n+1) n \sqrt[n]{(n+1)!}.$$

So the job of essential manifolds is to be the domain where Gromov's theorem holds. As far as I know, it is not so clear whether there exist larger classes of manifolds that satisfy such a systolic inequality.

The theorem is a generalisation of a theorem on tori:
Theorem (Loewner)Let $\gamma$ be a shortest closed geodesic in a flat torus $T^n$. Then
$$sys_1T^n = length(\gamma) \leq C_n \sqrt[n]{Vol(T^n)}.$$
Let $M$ be a 2-torus (with arbitrary metric), then
$$sys_1M \leq C_2 \sqrt{Area(M)}$$ and $C_2 = \sqrt{\frac{2}{\sqrt{3}}}$.
The 2-torus realising equality in this inequality is the quotient of $\mathbb{R}^2$ by the hexagonal lattice spanned by the 3rd roots of unity.

Pu proved a similar systolic inequality on $\mathbb{R}P^2$, so it is very reasonable to look for a class of closed manifolds that contain tori and real projective space and furthermore allow systolic inequalities.

Well, that's enough for today!