2000/03/22 by Weiping Li, Wei-Ping Li, Li, Weiping
Computer Science · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.DG #math.GT
paper · pdf · doi:10.48550/arxiv.math/0003134
20 pages, AMSLaTeX
arxiv created 2000/03/22 · openalex publication_date 2000/03/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To an integral homology 3-sphere Y, we assign a well-defined \Z-graded (monopole) homology MH_*(Y, I\e(\T; \e0)) whose construction in principle follows from the instanton Floer theory with the dependence of the spectral flow I\e(\T; \e0), where \T is the unique U(1)-reducible monopole of the Seiberg-Witten equation on Y and \e0 is a reference perturbation datum. The definition uses the moduli space of monopoles on Y \x \R introduced by Seiberg-Witten in studying smooth 4-manifolds. We show that the monopole homology MH_*(Y, I\e(\T; \e0)) is invariant among Riemannian metrics with same I\e(\T; \e0). This provides a chamber-like structure for the monopole homology of integral homology 3-spheres. The assigned function MHSWF: \I\e(\T; \e0)\ → \MH_*(Y, I\e(\T; \e0))\ is a topological invariant (as Seiberg-Witten-Floer Theory).