Subject: RE: four postings
From: "Ron Umble"
Date: Mon, 19 Feb 2007 11:34:47 -0500
Jim,
In his talk at the Lehigh conference last summer, Ralph Cohen discussed
the case [S,X] where S is a surface of genus g and X a 1-connected mfld.
You might want to check with him. He can probably point you in the
right direction.
Ron
Subject: question for all
From: jim stasheff
Date: Sat, 17 Feb 2007 16:04:33 -0500
I have a dim memory of someone's paper on homotopy
classes of maps FROM a torus
I have in mind the unstable range (S1)^n \to S2
with n > 2 so do not have the usual cohomotopy *group*
structure
anyone have a similar memory?
jim
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Subject: Re: four postings
From: "Ronnie Brown"
Date: Mon, 19 Feb 2007 17:24:59 -0000
Jim,
Maybe you are thinking of a paper by RHFox `Torus homotopy groups' Annals
of Math.
I was told by Brian Griffiths that Fox was really thinking of things like
Higher Homotopy van Kampen theorems (adding relations to homotopy groups)
but he dod not achieve this. He was still thinking in terms of groups,
whereas you all know my view that you have to use groupoids and higher
groupoids to get good local-to-global theorems, since in the cubical type
situation you can get algebraic inverses to subdivision. That was the
basic intuition pursued since 1966.
I also wondered why Frank Adam's arguments for the cellular appproximation
theorem (subdivision, and reproduced in my book) did not produce algebraic
results, as did the analogous 1-dimensional case. There seemed to be a
lack of an appropriate algebraic gadget. It took only 11 years to produce
it, fortunately collaborating with Chris Spencer, and then Philip Higgins.
I have put a number of papers as pdf files from
http://www.bangor.ac.uk/~mas010/publicfull.htm
They need some reorganisation into themes, perhaps. Will do in time.
Ronnie Brown