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

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