[ALGTOP-L] RE: ALGTOP-L Digest, Vol 136, Issue 1

Wu Jie matwuj at nus.edu.sg
Tue Oct 14 00:31:15 EDT 2008


I may want to add few comments to Ronie Brown's post for the referee's views on his paper `the interest is in the group and not in the module'.

First I would like to introduce a recent result of Graham Ellis and Roman Mikhailov. This result may be regarded as n-d van Kampen Theorem that generalizes Brown-Loday's theorem. The result also implies our group theoretic description of \pi_n(S^2). A brief description is as follows: 

Previously we have been to start with a space, say S^2, and we wanted to know what's going on for \pi_n(S^2). So we looked at the free group F_{n-1} of rank n-1 and considered n particular elements in F_{n-1}. Each element generates a normal subgroup of F_{n-1} (by taking its normal closure). Then we obtain n normal subgroups inside F_{n-1}. Now \pi_n(S^2) is simply given by the intersection of these n normal subgroups modulo the (iterated) commutator subgroup of these n normal subgroups.

Now Graham and Roman started with any group G and n normal subgroups inside G. Then one can construct a space X, given by the homotopy colimit of the n-cubical diagram of K(\pi,1)-spaces, with \pi_n(X) given by the intersection of these n normal subgroups modulo something close to the (iterated) commutator subgroup. 

There have been many interesting constructions of spaces from K(\pi,1)'s such as Quillen's plus construction. An interesting point of Ellis-Mikhailov's theorem is that the homotopy group measures certain canonical sub-quotient of a given group. Namely one can see "how the homotopy group is related to the original group".

Let us assume that we want to know what's going on for spheres. Well S^2 is the first example that we do not know \pi_n(S^2). In such a case, we may want to look at some particular groups. The good candidates are braid groups as the plus construction gives \Omega^2S^2. But these classical constructions do not tell "how to read out \pi_n(S^2) from the braid groups". So we looked at simplicial structure on braids. From this way, we can get "braided meaning" for \pi_n(S^2). One statement is that \pi_n(S^2) is given by (n+1)-strand Brunnian braids over S^2 modulo (n+1)-strand Brunnian braids over the disk for n>3. Also a connection between Vasiliev invariants and the homotopy was given in an unpublished paper of Fred Cohen and me. In addition, there is a connection between the homotopy groups and the word problem on certain canonical factor groups of the braid groups.

Next let me give some naïve/silly views on homotopy groups.

I assume that the homotopy groups are God-given. So we want to compute them. Then we found out that this is a non-trivial homework from God. On the other hand we found out that the elements in homotopy groups may often have particular meanings. For instance, \pi_3(S^2) has more information than integers as it contains the Hopf map. This may remind us that "the structure/properties on homotopy groups" is equally important as "the group".

I would consider "module" as a structure on a "group". The mod p homology groups are not just vector spaces. They are modules over the Steenrod algebra. 

The structure may also refer to many things. One could make products on homotopy, say Whitehead products and Toda brackets. One could get some special elements from the homotopy, say periodic elements as these elements have special meanings. One could look at the growth of the exponents on homotopy. Cohen-Moore-Neisendorfer told us that p^n would kill off all elements in the p-torsion component of \pi_*(S^{2n+1}) for an odd prime p. One could make resolutions on homotopy, say Goodwillie towers. Assume that we are interested in homotopy groups of finite complexes. The loop spaces of finite complexes are normally very big and so one could try to do representation theory for decomposing the spaces (like cutting a cake into smaller pieces). From simplicial view, one could do homotopy theory for "discrete objects" and then the homotopy groups become the derived groups for simplicial groups. This already gives a lot of group theoretic meanings for the homotopy groups. The group theoretic descriptions then give some kind of "combinatorial structure on homotopy". Say here is a group with finitely many generators and finitely many explicit defining relations whose center is just the homotopy; or one could consider certain action on certain explicit group whose fixed-point set is exactly the homotopy.

For avoiding the confusions of the referees, somehow it would be better to make more sections on homotopy groups in the Math Subject Classification of AMS. Namely the first question to a referee might be something like: is the author going to put some structures on the homotopy or to compute the group?

Best Regards,

Jie


-----Original Message-----
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Subject: ALGTOP-L Digest, Vol 136, Issue 1

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Today's Topics:

   1. Re: Detecting weak homotopy equivalences (R Brown)
   2. Cocycles (Christoph Wockel)


----------------------------------------------------------------------

Message: 1
Date: Sat, 13 Sep 2008 15:13:07 +0100
From: "R Brown" <ronnie.profbrown at btinternet.com>
Subject: Re: [ALGTOP-L] Detecting weak homotopy equivalences
To: <algtop-l at lists.lehigh.edu>
Message-ID: <231A4724C0034BD18F1E70682445A24D at RONNIENEW>
Content-Type: text/plain; format=flowed; charset="iso-8859-1";
	reply-type=original

The question of Yuli and the answers show the importance (standard of 
course) of the base point for these  homotopy invariants as tools. I want to 
make  a kind of philosophical point on this.

The base point for homotopy groups in a way implies the operations of the 
fundamental group on these groups. Mind you I once submitted  a paper using 
the 2-d van Kampen theorem to calculate the module \pi_2 in a particular 
case of a mapping cone, but that referee stated firmly that `the interest is 
in the group and not in the module'! Is that a general view? (The paper got 
published elsewhere.)

The operation may seem an encumbrance, but it seems needed  when one takes 
more than one base point, say a set A of base points, so obtaining an 
operation of the groupoid \pi_1(X,A) on the bundle of groups \pi_n(X,A): one 
wants to record how all these groups are  related.

This algebra can be used for example to describe the morphism of modules 
over groupoids
\pi_n(X,A) \to \pi_n(X/A,*)
when X= S^n \vee [0,1] and A={0,1}. (yes, you can the latter module in other 
ways).

So, as I have argued for over 40 years, it seems there are advantages in not 
being chained to one base point (but we need at least one, as the previous 
correspondence shows)

My question then is: what would that do to the theory of infinite loop 
spaces, and related areas? I think I have not the energy to attempt this 
rewriting, though I suspect I have some of the potential tools! It will 
perhaps seem a crazy idea!

Ronnie

>>>Missatge de rudyak <rudyak at math.ufl.edu>:
>>>
>>>>Don,
>>>>
>>>>A question for the list.
>>>>
>>>>Does somebody remember a reference to the following:
>>>>
>>>>The Whitehead theorem fails for non-pointed CW spaces, i.e. if a
>>>>map $f: X \to Y$ induces bijections $[S^n, X] \to [S^n,Y]$
>>>>(non-pointed homotopy classes), then $f$ is NOT necessarily a
>>>>(weak) homotopy equivalence.
>>>>
>>>>Moreover if I remember correctly, there is no a SET $S$ such that
>>>>the bijectivity   $[K, X] \to [K,Y]$ for all $K\in S$  implies
>>>>that $f$ is a homotopy equivalence.
>>>>
>>>>Yuli


------------------------------

Message: 2
Date: Sat, 13 Sep 2008 19:32:00 +0200
From: Christoph Wockel <christoph at wockel.eu>
Subject: [ALGTOP-L] Cocycles
To: algtop-l at lists.lehigh.edu
Message-ID: <E44111E3-3794-4BF7-AB04-B751E80AC1F6 at wockel.eu>
Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes

Hi,

I am searching for a reference for a generalisation of the following  
canonical cocycle to higher dimensions. Let G be a connected (locally  
contractible) top. group and take a section a of the evaluation map  
ev : PG \to G, continuous on a unit neighbourhood. Then (g,h)\mapsto  
[a(g)+g.a(h)-a(gh)] is a \pi_1(G)-valued group 2-cocycle on G,  
describing the universal cover of G as a central extension. Moreover,  
this cocycle is universal for discrete groups.

A similar construction works in higher dimensions, yielding for each  
(n-1)-connected group G a \pi_n(G) valued group (n+1)-cocycle.  
Moreover, this cocycle is universal for discrete groups. I've been  
searching for a reference for this, but did not succeed.

Thanks for any hints,

Christoph


------------------------------

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