Date: Mon, 1 Feb 1999 12:45:05 +0000 (GMT) From: "T.Porter" Subject: Re: Simpson on htpy gps Carlos has raised various excellent new points, but I feel that there is something that is slipping through the `fingers' of the discussion. The homotopy groups of complexes are only a pale reflection of their homotopy types them elves. The models for segments of the homotopy types of a complex (for instance the n-type for some relatively low value of n) contain the information on the homotopy groups in the range from 1 to n, but also a lot more. The van Kampen theorems that Ronnie Brown worked on with Higgins and then Loday attempted to provide ways of calculating the n-type, but finding GOOD models of the n-type is very hard and GOOD is NOT well defined! Various criteria can be used and calculability might be one. Hans Baues has put forward very powerful arguments in his work for small algebraic models of n-types but beyond the 3-type these are very difficult to define. Similarly if the complexes have just two non-trivial homotopy groups then very powerful techniques are available to classify their homotopy type. So algebraic models for homotopy types or `parts' of them at least should occupy the attention not just homotopy groups themselves. Another thought that haunts me is that although the fundamental group has several important geometric interpretations (notably through the theory of covering spaces) higher homotopy groups do not have any analogous property (at least as simply stated). Carlos has worked in this area I know but can I have the reaction of others on the list? For instance:What should the second `order' analogue of the covering space theory be? Any offers! Please note: Grothendieck's `Pursuing Stacks' took 654 pages (approx) and did not claim to have solved the problem! I'm sure Don would prefer that replies took less space than that! Tim Porter.