Unitary embeddings of finite loop spaces

AbstractIn this paper we construct faithful representations of saturated fusion systems over discrete p-toral groups and use them to find conditions that guarantee the existence of unitary embeddings of p-local compact groups. These conditions hold for the Clark–Ewing and Aguadé–Zabrodsky p-compact groups. We also show the existence of unitary embeddings of finite loop spaces.

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Pairings of {$p$}-compact groups and {$H$}-structures on the classifying spaces of finite loop spaces

Publications of the Research Institute for Mathematical Sciences ◽

10.2977/prims/1195144423 ◽

1998 ◽

Vol 34 (6) ◽

pp. 567-578 ◽

Cited By ~ 1

Author(s):

Kenshi Ishiguro

Keyword(s):

Compact Groups ◽

Classifying Spaces ◽

Loop Spaces ◽

Finite Loop

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RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

Glasgow Mathematical Journal ◽

10.1017/s0017089520000658 ◽

2021 ◽

pp. 1-29

Author(s):

DREW HEARD

Keyword(s):

Lie Group ◽

Loop Space ◽

Algebraic Model ◽

Compact Lie Group ◽

Loop Spaces ◽

Local Systems ◽

Special Case ◽

Finite Loop ◽

Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

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Classifying Spaces of Compact Lie Groups and Finite Loop Spaces

Handbook of Algebraic Topology ◽

10.1016/b978-044481779-2/50022-5 ◽

1995 ◽

pp. 1049-1094 ◽

Cited By ~ 3

Author(s):

D. Notbohm

Keyword(s):

Lie Groups ◽

Classifying Spaces ◽

Compact Lie Groups ◽

Loop Spaces ◽

Finite Loop

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Freudenthal's theorem and spherical classes in H*QS0

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091519000373 ◽

2019 ◽

Vol 63 (2) ◽

pp. 323-341 ◽

Cited By ~ 1

Author(s):

Hadi Zare

Keyword(s):

Special Focus ◽

Loop Spaces ◽

Finite Loop

AbstractThis note is on spherical classes in $H_*(QS^0;k)$ when $k=\mathbb{Z}, \mathbb{Z}/p$, with a special focus on the case of p=2 related to the Curtis conjecture. We apply Freudenthal's theorem to prove a vanishing result for the unstable Hurewicz image of elements in ${\pi _*^s}$ that factor through certain finite spectra. After either p-localization or p-completion, this immediately implies that elements of well-known infinite families in ${_p\pi _*^s}$, such as Mahowaldean families, map trivially under the unstable Hurewicz homomorphism ${_p\pi _*^s}\simeq {_p\pi _*}QS^0\to H_*(QS^0;\mathbb{Z} /p)$. We also observe that the image of the submodule of decomposable elements under the integral unstable Hurewicz homomorphism $\pi _*^s\simeq \pi _*QS^0\to H_*(QS^0;\mathbb{Z} )$ is given by $\mathbb{Z} \{h(\eta ^2),h(\nu ^2),h(\sigma ^2)\}$. We apply the latter to completely determine spherical classes in $H_*(\Omega ^dS^{n+d};\mathbb{Z} /2)$ for certain values of n>0 and d>0; this verifies Eccles' conjecture on spherical classes in $H_*QS^n$, n>0, on finite loop spaces associated with spheres.

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