2024/05/10 by Daniel Kasprowski, Kasprowski, Daniel, John William Nicholson +3
Computer Science · Mathematics · #19J25 #57K40 #57N65 #57Q10 #57R67 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · doi:10.48550/arxiv.2405.06637
openalex publication_date 2024/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Kreck's modified surgery gives an approach to classifying smooth 2n-manifolds up to stable diffeomorphism, i.e. up to connected sum with copies of Sn × Sn. In dimension 4, we use a combination of modified and classical surgery to study various stable equivalence relations which we compare to stable diffeomorphism. Most importantly, we consider homotopy equivalence up to stabilisation with copies of S2 × S2. As an application, we show that closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic. We give analogues of the cancellation theorems of Hambleton--Kreck for stable homeomorphism for homotopy up to stabilisations. Finally, we give a complete algebraic obstruction to the existence of closed smooth 4-manifolds which are homotopy equivalent but not simple homotopy equivalent up to connected sum with S2 × S2.