Date: Mon, 18 Jan 1999 13:04:53 +0000 (GMT)
From: Ronnie Brown <r.brown@bangor.ac.uk>
Subject: Re: question abt basepts

Tony's question can be put in the more general situation of homotopy
equivalence of pairs. The HEP is not invariant under homotopy equivalence of
pairs, but the WHEP is (WHEP means homotopies which are constant on the first
half extend). You will find some exercises on various HEPs in my book (1968
edition, as well as the later one, see p261 of the second edition), and this
has been further worked on by Rudger Knieboom.

The WHEP is of course a dualisation of a notion introduced by Dold.
I told Puppe about in the early 1960s, and he found a nice local
characterisation. (Archiv d. Math 18 (1967) 81-88). There are also basic
papers of Strom (Math Scand  19 (1966) 11-14, 22 (1969) 130-142).

So you can choose the definition of non degenerate base point so that it
is invariant under based homotopy equivalence.

There is also the nice result that a homotopy equivalence f of spaces with
non degenerate based points is a homotopy equivalence of spaces with base
point (if f preserves the base point). This also generalises to pairs.

Ronnie Brown

Just seen the related remarks by Gerd Laures on h-cofibrations.


On Sat, 16 Jan 1999, DON DAVIS wrote:

> Date: Sat, 16 Jan 1999 17:54:36 -0500 (EST)
> From: Tony Elmendorf <aelmendo@math.purdue.edu>
> Subject: degenerate basepoints
>
> Dear Colleagues:
>
> Let X and Y be based spaces which are based homotopy equivalent, and suppose
> X has a nondegenerate basepoint.  Prove or give a counterexample: Y must also
> have a nondegenerate basepoint.  (I know some smart person out there knows
> the answer. . .)
>
> Tony Elmendorf
>