Homotopical perspective on the long exact sequence in homology and Mayer-Vietoris

Question

For a while I have been wondering whether the long exact sequence in homology and the Mayer-Vietoris sequence can be phrased in homotopical terms. Recently, I heard that both may be reformulated via homotopy pullbacks/pushouts.

I am looking for a reference that develops this viewpoint, or at least mentions it in detail. I have not studied enough algebraic topology to fill in the blanks myself, nor have I worked enough with homotopy (co)limits to feel comfortable with them.

I have seen a couple of MSE answers which poke what I am looking for, but I cannot complete the picture myself.

Answer 1

Let $f : A \to X$ be a based map of based spaces. The homotopy pushout $X \coprod_A \text{pt}$ is called the homotopy cofiber, cofiber, or mapping cone of $f$; I'll denote it by $X/A$. Iterating this construction produces the cofiber sequence or Puppe sequence

$$A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma X/A \to \dots$$

which is in some sense the ancestor of all long exact sequences for relative homology and cohomology, although it's easier at this point to describe how to get the long exact sequence for relative cohomology. If $Z$ is another based space, then taking spaces of maps into $Z$ turns the cofiber sequence into a fiber sequence

$$[A, Z] \leftarrow [X, Z] \leftarrow [X/A, Z] \leftarrow [A, \Omega Z] \leftarrow [X, \Omega Z] \leftarrow [X/A, \Omega Z] \leftarrow \dots$$

which is built out of taking homotopy pullbacks in the same way that the cofiber sequence is built out of taking homotopy pushouts. If $Z$ is an Eilenberg-MacLane space $B^n G = K(G, n)$, taking $\pi_0$ of this fiber sequence produces the long exact sequence in relative (reduced) cohomology up to degree $n$, and taking $n \to \infty$ and piecing together the results gives the whole thing.

This is discussed a little in May's Concise Course in Algebraic Topology and a lot in Strom's Modern Classical Homotopy Theory.

Answer 2

I am not sure if this exactly what you want, but a homotopical perspective on homology is given in the book partially titled Nonabelian Algebraic Topology, EMS Tracts, vol 15 (2011) (pdf available there). The main results do not assume singular homology, but nevertheless give results such as the Relative Hurewicz Theorem, (!), and results on second relative homotopy groups not available by traditional methods, since these groups are usually non abelian. The book has comments on history and intuitions.

A major problem with homotopy theory is that for spaces, identifications in low dimensions have homotopical implications in high dimensions. To model this algebraically, one looks for algebraic objects with structure in a range of dimensions. These can of course be complicated, and to get then we need to move from spaces to spaces with structure, such as filtered spaces, as in this book; but they do allow for tools reflecting to some extent the homotopy of pushouts of such spaces with structure.

For an introduction to the background to these ideas, see this Dec 2014 presentation.