Bordism and resolution of singularities

TL;DR

This paper adapts algorithms for resolving singularities in complex algebraic varieties to demonstrate splitting of homology theories in complex bordism, with applications in symplectic topology and Hamiltonian fibrations.

Contribution

It introduces new methods to split homology theories from complex bordism to derived orbifolds and equivariant bordism, impacting symplectic topology and moduli space analysis.

Findings

Splitting of homology theories from complex bordism to derived orbifolds.

Definition of complex cobordism-valued Gromov-Witten invariants for all symplectic manifolds.

Constraints on the topology of Hamiltonian fibrations over S^2.

Abstract

We adapt algorithms for resolving the singularities of complex algebraic varieties to prove that the natural map of homology theories from complex bordism to the bordism theory of complex derived orbifolds splits. In equivariant stable homotopy theory, our techniques yield a splitting of homology theories for the map from bordism to the equivariant bordism theory of a finite group $\Gamma$, given by assigning to a manifold its product with $\Gamma$. In symplectic topology, and using recent work of Abouzaid-McLean-Smith and Hirschi-Swaminathan, we conclude that one can define complex cobordism-valued Gromov-Witten invariant for arbitrary (closed) symplectic manifolds. We apply our results to constrain the topology of the space of Hamiltonian fibrations over $S^2$. The methods we develop apply to normally complex orbifolds, and will hence lead to applications in symplectic topology that rely on moduli spaces of holomorphic curves with Lagrangian boundary conditions.