Flowing from intersection product to cup product

2021/06/10 by Greg Friedman, Friedman, Greg, Anibal M. Medina-Mardones +3
Mathematics · #55N45 #57R19 #57R25 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2106.05986

openalex publication_date 2021/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use a vector field flow defined through a cubulation of a closed manifold to reconcile the partially defined commutative product on geometric cochains with the standard cup product on cubical cochains, which is fully defined and commutative only up to coherent homotopies. The interplay between intersection and cup product dates back to the beginnings of homology theory, but, to our knowledge, this result is the first to give an explicit cochain level comparison between these approaches.

Citations

• Pseudohomology and homology
• Some new homology and cohomology theories of manifolds and orbifolds
• An axiomatic characterization of Steenrod's cup-i products

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• Alternative (Oriented) Singular Cochains and the Modified Cup Product
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