Framed configuration spaces and exotic spheres

2025/09/23 by Krannich, Manuel, Kupers, Alexander, Mezher, Fadi
#11F06 #18F50 #20E26 #55R80 #57R40 #57R60 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2509.19074

Abstract

We determine when an exotic sphere Σ of dimension d\not≡ 1\pmod4 can be detected through the homotopy type of its truncated Disc-presheaf. The latter records the diagram of framed configuration spaces of bounded cardinality in Σ with natural point-forgetting and -splitting maps between them. Our proof involves three ingredients that could be of independent interest: a gluing result for Disc-presheaves of manifolds divided into two codimension zero submanifolds, a version of Atiyah duality in the context of Disc-presheaves, and a computation of the finite residual of the mapping class group of the connected sums \sharp^g(S^2k+1× S^2k+1).

Citations

• Pontryagin-Weiss classes and a rational decomposition of spaces of homeomorphisms
• Residual finiteness of some automorphism groups of high dimensional manifolds
• ∞-operadic foundations for embedding calculus
• Simple homotopy invariance of the loop coproduct
• Reducibility of low dimensional Poincaré duality spaces
• Higher Topos Theory
• Configuration spaces are not homotopy invariant
• Manifold theoretic compactifications of configuration spaces

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