# TTT73: Warwick

Warwick Mathematics Institute, University of Warwick, Coventry.

## Date

23 March 2010

## Speakers

### John Greenlees

Homotopy invariant notions of complete intersections in algebra and topology

Homotopy invariant notions of complete intersections in algebra and topology

The talk will consider three well-known characterisations of the notion of complete intersection in commutative algebra and construct homotopy invariant versions of each. This allows them to be applied to homotopy theory (eg C*(X) for spaces X). It turns out these coincide (for classical rings) with the usual notions, and they coincide with each other in rational homotopy theory. The relation between them in characteristic p is under investigation. Joint work with D J Benson, K Hess, S Shamir.

### Oscar Randal-Williams

Resolutions of moduli spaces

Resolutions of moduli spaces

I will explain how to approximate moduli spaces of topological surfaces of genus g (possibly equipped with a tangential structure) by moduli spaces of strictly less genus, and how this often implies that the homology of these moduli spaces stabilises with g. This extends the famous stability theorem of J. Harer on the homology of the oriented mapping class groups to new families of mapping class groups preserving a tangential structure. I will discuss some new examples if there is time.

### Andrew Ranicki

The algebraic eta invariant

The algebraic eta invariant

The signature a 4k-dimensional Riemannian manifold with boundary (P,\partial P) was shown by Atiyah, Patodi and Singer (1973) to be the sum of the Hirzebruch L-genus and the \eta-invariant of \partial P. Many authors (Neumann, Meyer, Cappell-Lee-Miller, Bunke, Nemethi, ...) have subsequently given a more algebraic treatment of the \eta-invariant, defining it in particular for a closed (4k-1)-dimensional manifold N with a separating hypersurface M \subset N and a complex structure J on H^{2k-1}(M).

The talk will describe an even more algebraic treatment of the \eta-invariant, using notions of the algebraic theory of surgery. The algebraic \eta-invariant is related to the Wall non-additivity of the signature, the Maslov index, the Witt group of the function field of the reals, the Witt group of symplectic automorphisms, etc.

The von Neumann \rho-invariant of a knot is the high-dimensional knot concordance invariant defined by the average of the Tristram-Levine knot signatures. It plays an important role in the Cochran-Orr-Teichner calculations of the classical knot concordance group. The von Neumann \rho-invariant turns out to be the sum of the algebraic \eta-invariant and half the signature of the knot.

## Further information

TTT meetings are partially supported by an LMS Scheme 3 grant.