Characteristic numbers and group actions on homology spheres

Sung Sook Kim, Purdue University

Abstract

Let p and q denote odd prime numbers. In this thesis, we construct a smooth $\doubz \sb{p}$-action on a $\doubz \sb{q}$-homology sphere whose fixed point sets have nonzero Pontryagin numbers in dimension 4 and higher dimensions, and the following results are obtained. Theorem A. Let G be a cyclic group of order p, where p is an odd prime, and let q $\not=$ p be another odd prime. Then there is a smooth G-action on some $\doubz\sb{q}$-homology sphere such that the fixed point set is a closed connected 4-dimensional manifold with nonzero Pontryagin number. In fact, there are subgroups $Fix\sbsp{4}{p,q}$ of the oriented bordism group $\Omega\sb4$ such that: (i) every element of $Fix\sbsp{4}{p,q}$ contains a representative that is the fixed point set of some smooth G-action on some $\doubz\sb{q}$-homology sphere, (ii) $Fix\sbsp{4}{p,q}$ $\otimes$ $\doubq$ = $\Omega\sbsp{4}{SO}$ $\otimes$ $\doubq$. Theorem B. For each r $>$ 0 there is a smooth G-action on some $\doubz\sb{q}$-homology sphere such that the fixed point set is a closed connected 4r-dimensional manifold with nonzero Pontryagin numbers. In fact, there are subgroups $Fix\sbsp{4r}{p,q}$ of the oriented bordism group $\Omega\sb{4r}$ such that: (i) every element of $Fix\sbsp{4r}{p,q}$ contains a representative that is the fixed point set of some smooth G-action on some $\doubz\sb{q}$-homology sphere, (ii) $Fix\sbsp{4r}{p,q}$ $\otimes$ $\doubq$ = $\Omega\sbsp{4r}{SO}$ $\otimes$ $\doubq$, for all r $\geq$ 1 p $\not=$ q.

Degree

Ph.D.

Advisors

Schultz, Purdue University.

Subject Area

Mathematics

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