I'm looking for a good reference that has a detailed treatment of obstruction theory in the case where the target space is not simple. The specific situation I am interested in involves lifting a map of 3-skeletons from a $K(G, 1)$ to an arbitrary homotopy 3-type $X$ to the 4-skeletons (and hence to a true map between the spaces); the obstruction to this "should" live in $H^4(G, \pi_3(X))$ (with the appropriate action of $G$ on $\pi_3(X)$) via a local coefficient system, but I've been having trouble hashing out the details. Experts I've asked have given answers ranging from saying that it's impossible to saying that they're certain that it's possible but they don't know a reference that does it. Books I've looked at tend to gloss over the details, but they seem to indicate that this should work. Can anybody set me straight on this?

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    $\begingroup$ Trying to clarify what kind of obstruction you want. If $A$ and $B$ are $K(G,1)$ spaces with CW-decompositions, you have a map $f : A^3 \to B^3$ i.e. a map between the 3-skeleta. And you want to know if it extends to a map $f : A^4 \to B^4$ ? $\endgroup$ – Ryan Budney Jul 9 '10 at 2:56
  • $\begingroup$ My understanding is that A is a K(G,1) space and B is a space whose homotopy groups vanish above dimension 3. Suppose we have a map $f: A^3 \to B$. The problem then is to extend $f$ to a map from $A^4$ to $B$, and therefore from $A$ to $B$. $\endgroup$ – Gregory Arone Jul 9 '10 at 3:44
  • $\begingroup$ Sorry if this was unclear. The source space $A$ is $K(G, 1)$ (although this is probably inconsequential), and the target space $B$ (what I called $X$) is a connected homotopy 3-type (again, this is probably inconsequential). I want to extend a map between the 3-skeleta to the 4-skeleta, as you say, while possibly modifying the choice of 3-skeleton map (which itself was extended from the 2-skeleton in a non-unique way). It is this situation that, at least in the simple case, gives an obstruction in $H^4(A, \pi_3(B))$. $\endgroup$ – Evan Jenkins Jul 9 '10 at 3:48
  • $\begingroup$ Ah, okay, I hadn't heard the phrase "homotopy n-type" before. $\endgroup$ – Ryan Budney Jul 9 '10 at 5:46

Paul Olum developed some obstruction theory for maps into non-simple spaces back in the 1940-ies and 50-ies. You may want to check out his paper "Obstructions to extensions and homotopies", Annals of Mathematics, Vol 52, 1950, pp 1-50, if you have not looked at it yet.

  • $\begingroup$ Thank you! The approach taken in this paper seems much nicer than the less general and more mysterious treatment (due originally to Whitehead?) that one sees in more modern references. $\endgroup$ – Evan Jenkins Jul 13 '10 at 7:25

@Evans Jenkins: A comparison of the work of Whitehead and Olum is given in

Ellis, G.J. "Homotopy classification the J.H.C. Whitehead way". Exposition. Math. 6 (1988) 97--110.

He writes (and I leave the reader to find the citations):

``In view of the ease with which Whitehead's methods handle the classifications of Olum and Jajodia, it is surprising that the papers \cite{Olum53} and \cite{Jaj80} (both of which were written after the publication of \cite{W49:CHII}) make respectively no use, and so little use, of \cite{W49:CHII}.

``We note here that B. Schellenberg, who was a student of Olum, has rediscovered in \cite{Sch73} the main classification theorems of \cite{W49:CHII}. The paper \cite{Sch73} relies heavily on earlier work of Olum.''

Whitehead used what he calls "homotopy systems", which we now call "free crossed complexes"; the notion of crossed complex goes back to Blakers in 1948 (Annals of Math), and a full account is in the 2011 EMS Tract Vol 15 Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy grouopids. The relation between crossed complexes and chain complexes with operators is quite subtle; it was first developed by Whitehead, and in CHII he explains, in our terms, that crossed complexes have better realisation properties that chain complexes with operators. For example, the latter do not model homotopy 2-types.

Section 12.3 of the above Tract is on the homotopy classification of maps, including the non simply connected case, but it may be that your example is out of reach of the "linear" theory of crossed complexes. The homotopy classification of $3$-types requires quadratic information, see books by Baues and also

Ellis, G.J. "Crossed squares and combinatorial homotopy". Math. Z. (214} (1993) 93--110.

So there is sill a lot of work to be done!


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