A Counting Formula for the Kervaire Semi-Characteristic

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Ismayli, Behnam | 2011

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 42266 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Esfahani Zadeh, Mustafa
Abstract:
Similar to the classical Poincare-Hopf index formula for the Euler characteristic, Zhang gave a counting formula for the Kervaire semi-characteristic of a 4q+1 manifold. We first study Witten deformation techniques in the proof of the Poincare-Hopf formula. Then we will represent Zhang’s formula which uses similar analysis. Both these formulas establish a connection between topological and geometrical data on a manifold. Both proofs will be analytical. Index theory will play the ’bridge’ between the topology and analysis. We begin with an analytic interpretation of the topologically defined quantities as indices of differential operators. We deform the operators with respect to appropriate vector fields. The homotopy invariance property of index guarantees that the index will still be the characteristic we started with. On the other hand, the index of the deformed operator asymptotically concentrates around the singularities of the vector fields. The detailed calculation of this index in a neighborhood of the singularities will result in a proof of the formula for that characteristic
Keywords:
Witten Techniques ; Mod2 Index ; Kervaire Semi-Characteristic ; Analytical Interpretation

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