SU(n) and U(n) Representations of Three-Manifolds with Boundary

TL;DR

This paper develops new numerical invariants counting unitary and special unitary representations of 3-manifold groups extending from boundary subsurfaces, generalizing Casson's work and relating to Donaldson invariants.

Contribution

It introduces and computes new invariants for 3-manifolds that count representations extending boundary data, generalizing Casson's SU(2) invariants to U(n) and SU(n).

Findings

Invariants are independent of the boundary representation rho.

For T=0, invariants are computed up to sign.

For T>0, invariants form polynomials related to Donaldson invariants.

Abstract

Results are obtained on extending flat vector bundles or equivalently general representations from the fundamental group of S, a connected subsurface of the connected boundary of a compact, connected, oriented 3-dimensional manifold, to the whole manifold M. These are applied to representations of fundamental groups of 3-dimensional rational homology cobordisms. The proofs use the introduction and complete computation up to sign of new numerical invariants which "count with multiplicities and signs" the number of representations up to conjugacy of the fundamental group of M to the unitary group U(n) (resp., the special unitary group SU(n)) which when restricted to S are conjugate to a specified irreducible representation, rho, of the fundamental group of S. These invariants are inspired by Casson's work on SU(2) representations of closed manifolds. All the invariants treated here are independent of the choice of rho. If T equals the difference of the Euler characteristics of S and M and is non-negative, then a T times (dim U(n)) (resp., dim SU(n)) cycle is produced that carries information about the space of such U(n) (resp., SU(n)) representations. For T = 0, the above integer invariant results and it is entirely computed up to sign. For T > 0, under the assumption that rho sends each boundary component of S to the identity, a list of invariants for U(n) (resp. SU(n)) results which are expressed as a homogeneous polynomial in many variables, reminiscent of the work of Donaldson on 4-manifolds.