Equivalences of the form $Σ^V X \simeq Σ^W X$ in equivariant stable homotopy theory
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914860162351104 |
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| author | Balderrama, William |
| author_facet | Balderrama, William |
| contents | We study equivalences of the form $Σ^{V}X\simeq Σ^{W}X$, where $G$ is a compact Lie group, $X$ is a $G$-spectrum, and $V$ and $W$ are $G$-representations. These equivalences encode a periodicity phenomenon in $G$-equivariant homotopy theory which generalizes the classical James periodicity for $G = C_2$.
When $X = C(a_λ)$ is the cofiber of an Euler class, we construct an $RO(G)$-graded $J$-homomorphism $J\colon π_λKO_G\rightarrow π_\star^G C(a_λ)^\times$ which gives control over these periodicities. It also produces infinite periodic families in the $G$-equivariant stable stems. We illustrate this with several explicit examples.
More generally, our work gives information about $RO(G)$-graded units in equivariant stable cohomotopy rings. We apply this to construct universal periodicities and differentials in the $G$-homotopy fixed point spectral sequence, and other equivariant Atiyah--Hirzebruch spectral sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11000 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equivalences of the form $Σ^V X \simeq Σ^W X$ in equivariant stable homotopy theory Balderrama, William Algebraic Topology We study equivalences of the form $Σ^{V}X\simeq Σ^{W}X$, where $G$ is a compact Lie group, $X$ is a $G$-spectrum, and $V$ and $W$ are $G$-representations. These equivalences encode a periodicity phenomenon in $G$-equivariant homotopy theory which generalizes the classical James periodicity for $G = C_2$. When $X = C(a_λ)$ is the cofiber of an Euler class, we construct an $RO(G)$-graded $J$-homomorphism $J\colon π_λKO_G\rightarrow π_\star^G C(a_λ)^\times$ which gives control over these periodicities. It also produces infinite periodic families in the $G$-equivariant stable stems. We illustrate this with several explicit examples. More generally, our work gives information about $RO(G)$-graded units in equivariant stable cohomotopy rings. We apply this to construct universal periodicities and differentials in the $G$-homotopy fixed point spectral sequence, and other equivariant Atiyah--Hirzebruch spectral sequences. |
| title | Equivalences of the form $Σ^V X \simeq Σ^W X$ in equivariant stable homotopy theory |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2306.11000 |