2022/11/23 by Connor Malin, Malin, Connor · 1 citation
Mathematics · #57N65 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2211.12654
openalex publication_date 2022/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given smooth manifolds M and N, manifold calculus studies the space of embeddings Emb(M,N) via the "embedding tower", which is constructed using the homotopy theory of presheaves on M. The same theory allows us to study the stable homotopy type of Emb(M,N) via the "stable embedding tower". By analyzing cubes of framed configuration spaces, we prove that the layers of the stable embedding tower are tangential homotopy invariants of N. If M is framed, the moduli space of disks EM is intimately connected to both the stable and unstable embedding towers through the En operad. The action of En on EM induces an action of the Poisson operad poisn on the homology of configuration spaces H_*(F(M,-)). In order to study this action, we introduce the notion of Poincare-Koszul operads and modules and show that En and EM are examples. As an application, we compute the induced action of the Lie operad on H_*(F(M,-)) and show it is a homotopy invariant of M+.