An A_∞-coalgebra Structure on a Closed Compact Surface

2018/01/24 by Quinn Minnich, Minnich, Quinn, Ronald Umble +1
Mathematics · #57N05 #57N65 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1801.08071

openalex publication_date 2018/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P be an n-gon with n≥3. There is a formal combinatorial A_∞-coalgebra structure on cellular chains C_*(P) with non-vanishing higher order structure when n≥5. If X_g is a closed compact surface of genus g≥2 and P_g is a polygonal decomposition, the quotient map q:P_g→ X_g projects the formal A_∞-coalgebra structure on C_*(P_g) to a quotient structure on C_*(X_g), which persists to homology H_∗( X_g;ℤ₂) , whose operations are determined by the quotient map q, and whose higher order structure is non-trivial if and only if X_g is orientable or unorientable with g≥3. But whether or not the A_∞-coalgebra structure on homology observed here is topologically invariant is an open question.

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