Equivariant formality of istropy actions
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arXiv
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| Format: | Preprint |
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2015
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| _version_ | 1866909291860983808 |
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| author | Carlson, Jeffrey D. Fok, Chi-Kwong |
| author_facet | Carlson, Jeffrey D. Fok, Chi-Kwong |
| contents | Let $G$ be a compact connected Lie group and $K$ a connected Lie subgroup. In this paper, we collect an assortment of results on equivariant formality of the isotropy action of $K$ on $G/K$. If the isotropy action of $K$ on $G/K$ is equivariantly formal, then $G/K$ is formal in the sense of rational homotopy theory. This enables us to strengthen a theorem of Shiga--Takahashi to a characterization of equivariant formality in this case. Using a K-theoretic analogue of equivariant formality introduced and shown by the second-named author to be equivalent to equivariant formality in the usual sense, we provide a representation-theoretic characterization for equivariant formality of the isotropy action and give a new, uniform proof of equivariant formality for some classes of homogeneous spaces for which it was previously known. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1511_06228 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Equivariant formality of istropy actions Carlson, Jeffrey D. Fok, Chi-Kwong Algebraic Topology K-Theory and Homology 55N25, 55N15, 19L47, 57T15 Let $G$ be a compact connected Lie group and $K$ a connected Lie subgroup. In this paper, we collect an assortment of results on equivariant formality of the isotropy action of $K$ on $G/K$. If the isotropy action of $K$ on $G/K$ is equivariantly formal, then $G/K$ is formal in the sense of rational homotopy theory. This enables us to strengthen a theorem of Shiga--Takahashi to a characterization of equivariant formality in this case. Using a K-theoretic analogue of equivariant formality introduced and shown by the second-named author to be equivalent to equivariant formality in the usual sense, we provide a representation-theoretic characterization for equivariant formality of the isotropy action and give a new, uniform proof of equivariant formality for some classes of homogeneous spaces for which it was previously known. |
| title | Equivariant formality of istropy actions |
| topic | Algebraic Topology K-Theory and Homology 55N25, 55N15, 19L47, 57T15 |
| url | https://arxiv.org/abs/1511.06228 |