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Sutured manifolds and polynomial invariants from higher rank bundles

Aliakbar Daemi and Yi Xie

Geometry & Topology 24 (2020) 49–178

For each integer N 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.

sutured manifolds, higher rank Donaldson invariants, instanton Floer homology, Smith conjecture
Mathematical Subject Classification 2010
Primary: 57M27, 57R57, 57R58
Received: 4 March 2018
Revised: 25 March 2019
Accepted: 20 May 2019
Published: 25 March 2020
Proposed: Simon Donaldson
Seconded: Ciprian Manolescu, Peter Ozsváth
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