An algebraic model for the constant loops map

2024/11/27 by Luis Fernandez, Fernandez, Luis, Rivera, Manuel +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Chemistry · Computer Science · #16E40 #55P50 #57P10 #Algebraic Topology (math.AT) #FOS: Mathematics #Microtubule and mitosis dynamics #Molecular spectroscopy and chirality #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2411.18726

openalex publication_date 2024/11/27 · openalex created_date 2024/12/05 · openalex updated_date 2026/07/28

Abstract

For any simplicial complex X with a total ordering of its vertices, one can construct a chain complex \mathbbL_\bullet(X) generated by necklaces of simplices in X, which computes the homology of the free loop space of the geometric realization of X. Motivated by string topology, we describe two explicit chain maps C_\bullet(X) → \mathbbL_\bullet(X), where C_\bullet(X) denotes the simplicial chains in X, lifting the homology map induced by embedding points in |X| into constant loops in the free loop space of |X|. One of the maps has a convenient combinatorial description, while the other is described in terms of higher structure on C_\bullet(X).

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