Homological stability of moduli spaces of high-dimensional 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 1 Let , 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 2 The 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 3 Under 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 ).
-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.
Is the theorem true for all groups? Yes. We are able to estimate only in the polycyclic case.