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Title:
Spin Manifolds and Spin Bordism
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Abstract:
In this talk we will discuss Spin manifolds, especially those of dimensions 1,2, 3, and 4. Spin manifolds of dimension 1 and 2 are quite an interesting warm-up: They are related to immersed surfaces in 3-space and to quadratic forms on Z/2Z vector spaces and gauss sums. Spin manifolds of dimension 4 are closely related to quadratic forms on free abelian groups. The central focus on my recent work (joint with Greg Brumfiel) has been 3-dimensional spin bordism. For each topological space X there is the 3-dimensional spin bordism group of X. This is the group generated by all continuous maps of closed 3-dimensional spin manifolds into X modulo the relation that any boundary of a compact 4-dimensional Spin manifold mapping into X is set equal to zero. My interest in these groups was sparked by questions from Kapustin, inspired by work that he and Gaiotto were doing on topological phases of matter. Kapustin's question was to describe, in some explicit terms, the group of homomorphisms from the 3-dimensional spin bordism group to the circle. We give a complete answer to Kapustin's question.
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