Sutured manifolds and polynomial invariants from higher rank bundles

Daemi, Aliakbar; Xie, Yi

doi:10.2140/gt.2020.24.49

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Sutured manifolds and polynomial invariants from higher rank bundles

Aliakbar Daemi and Yi Xie

Geometry & Topology 24 (2020) 49–178

DOI: 10.2140/gt.2020.24.49

Abstract

For each integer $N \geq 2$ , Mariño and Moore defined generalized Donaldson invariants by the methods of quantum field theory, and made predictions about the values of these invariants. Subsequently, Kronheimer gave a rigorous definition of generalized Donaldson invariants using the moduli spaces of anti-self-dual connections on hermitian vector bundles of rank $N$ . We confirm the predictions of Mariño and Moore for simply connected elliptic surfaces without multiple fibers and certain surfaces of general type in the case that $N = 3$ . The primary motivation is to study $3$ –manifold instanton Floer homologies which are defined by higher rank bundles. In particular, the computation of the generalized Donaldson invariants is exploited to define a Floer homology theory for sutured $3$ –manifolds.

Keywords

sutured manifolds, higher rank Donaldson invariants, instanton Floer homology, Smith conjecture

Mathematical Subject Classification 2010

Primary: 57M27, 57R57, 57R58

References

Publication

Received: 4 March 2018

Revised: 25 March 2019

Accepted: 20 May 2019

Published: 25 March 2020

Proposed: Simon Donaldson

Seconded: Ciprian Manolescu, Peter Ozsváth

Authors

Aliakbar Daemi
Simons Center for Geometry and Physics State University of New York Stony Brook, NY United States

Yi Xie
Beijing International Center for Mathematical Research Peking University Beijing China

Volume 24, issue 1 (2020)