Localization of certain torus actions on odd-dimensional manifolds and its applications
Let torus $T$ act on a compact smooth manifold $M$, if the equivariant cohomology $H^_T(M)$ is a free module of $H^_T(pt)$, then according to the Chang-Skjelbred Lemma, $H^*_T(M)$ can be localized to the $1$-skeleton $M_1$ consisting of fixed points and $1$-dimensional orbits. Goresky, Kottwitz and...
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Online Access: | http://hdl.handle.net/2047/D20248836 |
Summary: | Let torus $T$ act on a compact smooth manifold $M$, if the equivariant cohomology $H^_T(M)$ is a free module of $H^_T(pt)$, then according to the Chang-Skjelbred Lemma, $H^*_T(M)$ can be localized to the $1$-skeleton $M_1$ consisting of fixed points and $1$-dimensional orbits. Goresky, Kottwitz and MacPherson considered the case where $M$ is an algebraic manifold and $M_1$ is $2$-dimensional, and introduced a graphic description of its equivariant cohomology. In this
thesis, firstly we will compute the equivariant cohomology of $3$-dimensional closed manifolds with circle actions, then we will give graphic descriptions of equivariant cohomology of certain class of torus actions on odd-dimensional manifolds with $3$-dimensional $1$-skeleton. |
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