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abstract: In the early 1940s, P. A. Smith showed that if a finite $$p$$-group $$G$$ acts on a finite dimensional complex $$X$$ that is mod $$p$$ acyclic, then its space of fixed points, $X^G$, will also be mod $$p$$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $$G$$-space $$X$$ is a equivariant homotopy retract of the $$p$$-localization of a based finite $$G$$-C.W. complex, given $H
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Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant
In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin ( 2 ) \operatorname {Pin}(2) -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the Pin ( 2 ) \operatorname {Pin}(2) -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from B Pin ( 2 ) B\operatorname {Pin}(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j j -based Atiyah–Hirzebruch spectral sequence.
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- PAR ID:
- 10324938
- Date Published:
- Journal Name:
- Communications of the American Mathematical Society
- Volume:
- 2
- Issue:
- 2
- ISSN:
- 2692-3688
- Page Range / eLocation ID:
- 22 to 132
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/cams/4
