Fixed point indices, transfers and path fields

Chen-Farng Benjamin, Purdue University

Abstract

Let $V$ be a vector field on a compact differentiable manifold $M$. Assume $V$ has no zeros on the boundary $\partial M$. Marston Morse discovered in 1929 an interesting formula relating the indices of the vector field $V$ and of a vector field $\partial\underline{\enspace} V$ defined on a part of the boundary to the Euler characteristic of $M:{\rm Ind}(V) + {\rm Ind} (\partial\underline{\enspace} V)$ = $\chi(M)$; this formula generalizes the well-known Poincare-Hopf Index Theorem for vector fields. In this work we obtain analogous formulas, $I(f)$ + $I(rf\vert\sb{\partial\underline{\enspace} M}$ = $\chi(M)$ for fixed point indices, $t\sb{rf}$ = $t\sb{f}\ \circ\ \breve i\sbsp{1}{\*}+t\sb{\partial\underline{\enspace} f}\ \circ\ \breve i\sp{\*}\ \circ\ \breve i\sbsp{2}{\*}$ for fixed point transfers and Ind($\sigma$) + Ind($\partial\underline{\enspace}\sigma$) = $\chi(M)$ for path field indices.

Degree

Ph.D.

Advisors

Gottlieb, Purdue University.

Subject Area

Mathematics

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