## Notes of Nina Friedrich’s Cambridge lecture 17-05-2017

Homological stability of moduli spaces of high-dimensional manifolds

1. Manifolds

Say a sequence manifolds and maps satisfies homological stability if induced maps on homomology groups become eventually isomorphisms.

How does one prove such a property? We are interested in the special case of diffeomorphim groups. Assume these manifolds are groups. Find a highly connected complex ${X}$ with action of ${X_n}$, then use a Serre spectral sequence.

Theorem 1 Let ${2n\geq 6}$, let ${W}$ be a ${2n}$-manifold with non-empty boundary, and with virtually polycyclic fundamental group. Consider the map

$\displaystyle \begin{array}{rcl} BDiff_\partial(W)\rightarrow BDiff_\partial(W\#(S^n\times S^n)) \end{array}$

induced by surging in ${(\partial W\times I)\#(S^n\times S^n)}$. Then this map is an isomorphism in homology ${H_k}$ for ${k}$ bounded above by ${\frac{1}{2}(g(W)-h(\pi_1(W))-5)}$.

Here, ${h}$ is the Hirsch length and ${g}$ the genus defined as follows:

Definition 2 The genus of a manifold is the maximum num of copies of ${S^n\times S^n}$ whose connected sum (with a disk deleted) can be embedded in ${W}$.

Let ${I^{tr}_n(W)}$ be the set of immersions of ${S^n}$ in ${W}$ with trivial normal bundle, together with a path from base point of W to image of base point of the sphere. This is the ${n}$-th homotopy group of ${n}$-frames in ${W}$. It has the structure of an abelian group. Furthermore, ${I^{tr}_n(W)}$ is a ${{\mathbb Z}[\pi_1(M)]}$-module.

The intersection form defines a ${{\mathbb Z}[\pi_1(M)]}$-valued bilinear form ${\lambda}$ on ${I^{tr}_n(W)}$. It counts a loop, obtained from paths from base-point, for each intersection point of spheres. It is skew-hermitian (note that the group ring has an involution).

Finally, there is a map ${\mu:I^{tr}_n(W)\rightarrow{\mathbb Z}[\pi_1(M)]/\Lambda}$ that counts Modding out by ${\Lambda=im(\lambda)}$ is necessary to make ${\mu}$ linear, since otherwise ${\mu(a+b)=\mu(a)+\mu(b)+\lambda(a,b)}$.

Equipped with ${\lambda,\mu}$, ${I^{tr}_n(W)}$ is a “quadratic module” in the sense of Wall.

A hyperbolic module is ${{\mathbb Z}[\pi_1(M)]^2}$ with basis ${(e,f)}$ satisfying ${\mu(e)=\mu(f)=0}$ and ${\lambda}$ has matrix ${\begin{pmatrix} 0 & 1 \\ (-1)^n & 0 \end{pmatrix}}$. This is the quadratic module of a punctured product of two spheres.

Define ${HU(M)}$ as the space of semi-simple quadratic modules mod those containing ${k+1}$ copies of a hyperbolic module.

Our topological theorem follows from the following algebraic result.

Theorem 3 Under the same assumptions as Theorem 1, ${HU(I^{tr}_n(W))}$ is ${\frac{1}{2}(g(W)-h(\pi_1(W))-6)}$-connected.

In fact our theorem is more general, involving an invariant of rings ${R}$, the unitary stable rank ${usr(R)}$, instead of Hirsch length, and an invariant of quadratic modules, ${g(M)}$, the Witt index of ${M}$, is the maximal number of copies of the hyperbolic module whose direct sum can be embedded in ${M}$.

Say a sequence ${v_i}$ in ${M}$ is unimodular, if there exist ${f_i:R\rightarrow M}$ and ${\phi_i:M\rightarrow R}$ such that ${f_i(1)=v_i}$ and ${\phi_j\circ f_i=\delta_i^j \,1_R}$. Say a ring ${R}$ satisfies ${(S_n)}$ if for every ${n}$-term unimodular sequence ${r_i}$, there exists scalars ${t_i\in R}$ such that ${(r_1+t_1r_{n+1},\ldots,r_n+t_nr_{n+1})}$ is again unimodular. Say ${R}$ has unitary stable rank ${n}$, ${usr(R)=n}$, if ${n}$ is minimal such that ${(S_n)}$ and ${(T_{n+1})}$ hold. I skip the definition of ${(T_n)}$.

Example. Fields have ${usr=1}$ and integers have ${usr=2}$ (the Euclide algorithm proves ${(S_2)}$).

3. Proof

${\lambda}$-unimodularity is the variant where linear forms ${\phi_i}$ are required to be of the form ${\lambda(w_i,\cdot)}$. For sums of hyperbolic modules, this makes no difference, but it does in general. This is the main new difficulty compared to previously existing results.

4. Question

Is the theorem true for all groups? Yes. We are able to estimate ${usr}$ only in the polycyclic case.