Equivariant Nerve Lemma, simplicial difference, and models for configuration spaces on simplicial complexes

Emilio J. González; Jesús González

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arXiv · 2004.09020

Equivariant Nerve Lemma, simplicial difference, and models for configuration spaces on simplicial complexes

Emilio J. González·Jesús González

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Abstract

Wiltshire-Gordon has introduced a homotopy model for ordered configuration spaces on a given simplicial complex. That author asserts that, after a suitable subdivision, his model also works for unordered configuration spaces. We supply details justifying Wiltshire-Gordon's assertion and, more importantly, uncover the equivariant properties of his more-general simplicial-difference model for the complement of a subcomplex inside a larger complex. This is achieved by proving an equivariant version of the Nerve Lemma. In addition, in the case of configuration spaces, we show that a slight variation of the model has better properties: it is regular and sits inside the configuration space as a strong and equivariant deformation retract. Our variant for the configuration-space model comes from a comparison, in the equivariant setting, between Wiltshire's simplicial difference and a well known model for the complement of a full subcomplex on a simplicial complex.

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BibTeX RIS

Emilio J. González, Jesús González. 2020-04-20. Equivariant Nerve Lemma, simplicial difference, and models for configuration spaces on simplicial complexes. https://arxiv.org/abs/2004.09020

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