** 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 with action of , then use a Serre spectral sequence.

Theorem 1Let , let be a -manifold with non-empty boundary, and with virtually polycyclic fundamental group. Consider the map

induced by surging in . Then this map is an isomorphism in homology for bounded above by .

Here, is the Hirsch length and the genus defined as follows:

Definition 2The genus of a manifold is the maximum num of copies of whose connected sum (with a disk deleted) can be embedded in .

**2. Quadratic modules **

Let be the set of immersions of in with trivial normal bundle, together with a path from base point of W to image of base point of the sphere. This is the -th homotopy group of -frames in . It has the structure of an abelian group. Furthermore, is a -module.

The intersection form defines a -valued bilinear form on . 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 that counts Modding out by is necessary to make linear, since otherwise .

Equipped with , is a “quadratic module” in the sense of Wall.

A hyperbolic module is with basis satisfying and has matrix . This is the quadratic module of a punctured product of two spheres.

Define as the space of semi-simple quadratic modules mod those containing copies of a hyperbolic module.

Our topological theorem follows from the following algebraic result.

Theorem 3Under the same assumptions as Theorem 1, is -connected.

In fact our theorem is more general, involving an invariant of rings , the *unitary stable rank* , instead of Hirsch length, and an invariant of quadratic modules, , the *Witt index* of , is the maximal number of copies of the hyperbolic module whose direct sum can be embedded in .

Say a sequence in is unimodular, if there exist and such that and . Say a ring satisfies if for every -term unimodular sequence , there exists scalars such that is again unimodular. Say has unitary stable rank , , if is minimal such that and hold. I skip the definition of .

**Example**. Fields have and integers have (the Euclide algorithm proves ).

**3. Proof **

-unimodularity is the variant where linear forms are required to be of the form . 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 only in the polycyclic case.