Homotopy Theory with Bornological Coarse Spaces

Slant products on the Higson–Roe exact sequence

“…In analogy to the first part of Lemma 4.12 we shall now prove the following theorem, which says that the external and slant products which we have constructed are compatible in the sense that (x × z)/θ = x × (z/θ). It would also be nice to have an analogue of the second part of Lemma 4.12, but this would require the construction of a secondary (16) external product of the form…”

Secondary cup and cap products in coarse geometry

“…Dual to the ordinary coarse cohomology, there is also an ordinary coarse homology HX * (−, −; M). In a special case, its definition has already been given in [23, page 453], and the general version was developed in [3,Section 6.3]. We refer to [21,Section 4.1] for proofs of basic properties, including that it is a coarse homology theory satisfying the strong excision axiom.…”

Injectivity results for coarse homology theories

“…We further recall the notion of an equivariant coarse homology theory, in particular its universal version with values in the stable ‐category of equivariant coarse motivic spectra. Most of this material has been developed in [13] (see also [10] for the nonequivariant case).…”

“…Remark In [10] we called this property strongly bounded geometry . It is not invariant under coarse equivalences.…”

“…In order to capture the large‐scale geometry of metric spaces like (with its word metric), we introduced the category of ‐bornological coarse spaces in [10, 13]. We further defined the notion of an equivariant coarse homology theory.…”