One posting: A response to Goodwillie's request for streamlined approaches to singular homology........DMD ________________________________________________________ Subject: Re: singular theory. From: Ronnie Brown Date: Mon, 22 Nov 2004 14:08:38 +0000 One of the questions is: what does one want from singular theory? A difficult part is the link with homotopy theory, leading to the Brouwer degree, and Hurewicz theorems, which seem to need the full homology theory and basic homotopy theory as well. So I'd like to point out the bypass using cubical methods and directly constructing homotopy invariants of a filtered space, started in my papers with Philip Higgins ``On the algebra of cubes'', {\em J. Pure Appl. Algebra} 21 (1981) 233-260. ``Colimit theorems for relative homotopy groups'', {\em J. Pure Appl. Algebra} 22 (1981) 11-41. In the space of 57 pages, and assuming little except general topology and some CW-complex theory, we get the relative Hurewicz theorem as a special case of a van Kampen type Theorem, yielding (for example) \pi_n(A \cup B, B) as an induced module (crossed if n=2) if (A,A \cap B) is (n-1)-connected. Jim Stasheff rightly remarked on the need of students for calculation. The above leads to specific calculations utilising the action of the fundamental group on higher relative homotopy groups, which are especially fun in the nonabelian case n=2. Also the student who wants definition-theorem-proof will get it in these papers (with examples). These methods give only a partial account of algebraic topology, many areas not having been worked on or out, e.g. Poincare duality. So that gives room for thesis topics! So it is still desirable for a student to understand the use of the Mayer-Vietoris sequence, as one example. Ronnie Brown ________________________