Matthew Stoffregen

Manolescu Invariants of Connected Sums

We give inequalities for the Manolescu invariants α,β,γ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furutas Theorem, the existence of a Z∞ subgroup of the homology cobordism group. To our knowledge, this is the first proof of Furutas Theorem using monopoles. We also provide information about Manolescu invariants of the connected sum of n copies of a three-manifold Y, for large n.

The mod two cohomology of the moduli space of rank two stable bundles on a surface and skew Schur polynomials

We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface usingmethods ofZagier. _e resulting formula is related to a generating function for certain skewSchur polynomials. As an application,we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under themapping class group action.

Klein-four connections and the Casson invariant for nontrivial admissible

Given a rank 2 hermitian bundle over a 3-manifold that is non-trivial admissible in the sense of Floer, one defines its Casson invariant as half the signed count of its projectively flat connections, suitably perturbed. We show that the 2-divisibility of this integer invariant is controlled in part by a formula involving the mod 2 cohomology ring of the 3-manifold. This formula counts flat connections on the induced adjoint bundle with Klein-four holonomy.