Index of transverse Dirac operator and cohomotopy Seiberg-Witten\n invariant for codimension 4 Riemannian foliation

2019/03/06 by Dexie Lin, Lin, Dexie
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1903.02391

openalex publication_date 2019/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For closed manifolds endowed with a Riemannian foliation of codimension 4,\none can define a transversal Seiberg-Witten map. We show that there is a finite\ndimensional approximation for such a map. By such a method and under the\ncondition that H¹_b(M)\∩ H¹(M, mathbb Z) is a lattice of H¹_b(M), we\ncan define a foliated version of Bauer-Furuta invariant. Moreover, if the basic\ncohomological group is of zero dimension, we can give an estimate for the index\nof transversal Dirac operator of a foliated spin structure. Furthermore, under\nthe condition that H^\±_b(M)=1, we show the vanishing of the index of the\ntransverse Dirac operator. This gives a topological condition for the vanishing\nof the index of the transverse Dirac operator.\n

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