Some Aspects on Coarse Homotopy Theory

Abstract

The most important examples of coarse spaces arise from proper metrics on spaces or from metrisable compactifications. In both cases, the bounded subsets are precisely the relatively compact ones. On the other hand, one of the characteristic properties a coarse map required to have is formulated by the bounded subsets. Motivated from the above two facts and in order to be able to develop the notion of locally properness which has been introduced by Viet-Trung Luu for discrete spaces, we introduce a new notion of compatibility between the coarse structure and the topology when a coarse space also carries a topology. The spirit of this is to let the local part of the theory govern by the topology of spaces. Having established this we are able to introduce some basic notions such as pull-back and push-forward coarse structures, and products and coproducts of coarse spaces which also are carrying a topology with the required compatibility between the topology and the coarse structure. We use the notion of basepoint projection introduced by Paul Mitchener and Thomas Schick to develop a notion of pointed coarse spaces which leads us to a new notion of collapsing from a coarse point of view. This enables us to introduce some essential notions such as coarse quotient spaces, coarse spaces obtained by coarse collapsing and coarse spaces obtained by coarse attaching via a coarse map. We introduce a new notion which in a sense is the analogue for coarse geometry of locally compactness for topology and investigate some of its properties. Then, we develop basic notions in the coarse homotopy theory including some constructions such as coarse smash product, coarse suspensions and coarse mapping cone needed to develop coarse homotopy theory and we prove some of their properties. The coarse homotopy groups are introduced next and then we develop an exact sequence of coarse homotopy groups. We also give a more complete exposition on the coarse CW-complexes broad enough to provide an appropriate foundation in order to carry over more of the tools from algebraic topology into coarse geometry. Having established that we prove a coarse version of the theorem of J.H.C. Whitehead which allows certain aspects of coarse homotopy classification. Next, we pursue a big step forward and calculate the coarse homotopy groups of the standard coarse spheres. More precisely, we prove $\pi^{crs}_k(S^n_{\preal}) \cong \pi_k(S^n)$ for all $k \leq n$. This has some intense applications, namely, this enables us to carry over some important theorems from algebraic topology concerning the coarse homotopy groups of coarse CW-complexes when they are $\preal$-spaces. Then, as a one result of these theorems, we introduce the coarse Eilenberg-Maclane spaces.