Re: covering spaces and groupoids

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

Dear Bill,

You ask:

In the applications of algebraic topology to topology, where do the 
‘basepoints’ originate?

I'd like to give a different type of answer to that suggested in your 
list. (yet another is : whoopee, we have a group!)

After I first thought of the van Kampen theorem for the whole 
fundamental groupoid I realised one wanted *computation*, and the whole 
fundamental groupoid of a space was too big for that. The fundamental 
group at a base point was too small in may cases, such as the circle. 
The solution that seemed right to me, and which took a while to find, 
was the fundamental groupoid on a set of base points chosen conveniently 
according to the geometry of a given situation. Eldon Dyer said I ought 
to take a hard line on this: if the connected space X is the union of 
127 open sets whose intersections have 3,272 components, you do not want 
to take a single base point! Such situations (even with infinitely many 
components) occur in group theory applications, and analogously in 
topology in connected covering spaces over a union of 2 open sets.

I advocated the fundamental groupoid on a set of base points in my 1968 
book `Elements of Modern Topology', but this concept has I think not 
been mentioned in any later date algebraic topology text in English by 
other authors. Also group theorists are happy with graphs and free 
groups, but are very wary of the free groupoid on a graph! The only new 
result from that book that has been taken up is the gluing theorem for 
homotopy equivalences, but its origin is usually unacknowledged.

Looking at the way this groupoid van Kampen theorem could be used, it 
seemed amazing to me that one could obtain *complete* information on a 
fundamental group by deducing that from knowledge of a larger structure, 
for which one had colimit information, whereas my tries with nonabelian 
cohomology gave only exact sequences. It seems that groupoids have the 
advantages of structure in dimensions 0 and 1, and that this is needed 
for what Grothendieck later called `integration of homotopy types'.

Could one find analogous objects with structure in dimensions 0, 1, 
...,n? This question led (after many years, and with fortunate 
collaborations) to higher dimensional van Kampen theorems, which gave 
quite new information on homotopy invariants, some of them nonabelian, 
e.g. second relative homotopy groups, triad homotopy groups, n-adic 
Hurewicz Theorems. Such results were published in 1978, 1981 (with 
Higgins), 1987 (with Loday). These theorems are, I think, not even 
mentioned in any texts on algebraic topology (except mine). Some current 
writers (Faria Martins, Kauffman, Ellis, Mikhailov,...) are using these 
techniques.

So for me the question is: why are people unwilling to throw off the 
shackles of a single base point? I would welcome enlightenment. It may 
be that it is found just too hard to fit this idea into what is 
currently considered the `real world', and so to obtain new results. For 
example, what happens to iterated loop space theory, with more than one 
base point?

Those interested in the sociology of science might like an excerpt from 
a lecture by Alan MacKay on icosahedral symmetry at the LMS in 1985. He 
said the reaction went through three phases:

Phase 1) It is false.

Phase 2) It is true, but unimportant.

Phase 3) It is true, it is very important; and we have known it for years!

All the best

Ronnie




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