The homotopy type of the Poincaré cobordism category for surfaces

2025/10/09 by Azélie Picot, Picot, Azélie
Mathematics · #57P10 (Secondary) #57R90 (Primary) 18F50 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2510.07659

openalex publication_date 2025/10/09 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

We define a version of the surface cobordism category Cob₂^SG(X) over a base space X where surfaces are considered up to self homotopy equivalences instead of diffeomorphisms. We prove the induced functor BCob₂^SG(-): S → S is not 1-excisive. We show its first derivative ∂₁ BCob₂^SG(-) in the Goodwillie sense is equivalent to a Thom spectrum over Bhaut_*⁺(S²).

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