Crossed module structure on homotopy groups

Question

A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy

• $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
• $cc'c^{-1}=(\partial c)\cdot c'$

Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, $$ \pi_2(X,A) \stackrel{\partial}{\rightarrow} \pi_1(A) $$ is a crossed module. Simply put, my question is: what does this give us, other than an extra bit of structure? Does knowing that this is true aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)

Answer 1

There are numerous calculations that are easily done with crossed module techniques that are much more difficult to obtain using `traditional' homotopy theory. Some of these use the next stage up, that is crossed squares, and the resulting non-Abelian tensor product. A neat sample calculation is of the homotopy type of the suspension of a K(G,1), if I remember rightly. This is given as the kernel of the commutator map from $G\otimes G$ to $G$.

You ask is it good at distinguishing spaces. The answer is most decidedly yes. (But I would say that wouldn't I.) MacLane and Whitehead proved that the crossed module models the homotopy 2-type, extending the classification of homotopy 1-types by groups. (Ok there is a price to pay. The correspondence gives 2-types correspond to equivalence classes of crossed modules but the equivalence relation is algebraic not topological in nature so that is reasonable.) Loday proved that homotopy n-types had algebraic models which were crossed n-cubes, n-fold generalisations of crossed modules.

The cool thing is that conceptually they are one of a linked set of models for low dimensional homotopy information that have geometric significance, and yet are relatively easy to manipulate. I like to say that a crossed module is a normal subgroup that is not a subgroup. Crossed modules 'are' also 2-groups, cat$^1$-groups and various other equivalent formulations.

For some higher dimensional vKT applications, look at higher Hopf formulae in work by Brown and Ellis.

For applications of crossed modules in non-Abelian cohomology etc. look at Larry Breen's work, or for a gentle introduction, my Menagerie notes which you can find on the n-Lab. I could go on listing things but will stop here. If you (or anyone else needs more detail) ask me or ask here.

Answer 2

1) The book "2D homotopy and combinatorial group theory" (1993) is decidedly oriented towards applications and problem solving, and does discuss crossed modules in Chapters 2 and 4, and further mentions them in Chapters 5, 9 and 11 (different chapters have different authors). There are some results proved using crossing modules and not involving crossed modules in the statement. Yet I cannot refrain from quoting some honest disclaimers:

By A. J. Sieradski (p. 75):

"The difficult nature of free crossed homotopy modules limits the applicability of the 2-dimensional homotopy classification [in terms of crossed modules]. The cellular chain complex of the universal coverings of two-dimensional complexes offers an abelianized version of the classification that is much more practical."

By W. A. Bogley (p. 311)

"Whitehead's work on crossed modules provides an abstract algebraic description of the second homotopy group of a 2-complex [Wh41$_1$, page 427]. Abelianizing, one obtains the homological description of $\pi_2$ in terms of Reidemeister chains [Re34, Re50, Wh46]. (See also Chapter II, Theorem 3.8, in this volume.) However, as Whitehead himself observes [Wh41$_1$, page 409], [Wh49$_2$, page 495], neither of these descriptions leads to effective general calculations of $\pi_2$. Nor do they shed any practical light on Whitehead's question on the heredity of asphericity."

2) D. Conduché, Question de Whitehead et modules précroisés (1996)

"The author turns [the Whitehead asphericity problem] into an algebraic question by showing that $\pi_2(Y)$ is the intersection of the terms of the lower central series of the crossed module $\pi_2(Y,Y^1)\to\pi_1(Y^1)$, where $Y^1$ is the 1-skeleton of $Y$."

3) J. Huebschmann, Braids and crossed modules (2009)

The main result of the present paper, Theorem 5.1 below, says that, as a crossed module over itself, the Artin braid group $B_n$ has a single generator, which can be taken to be any of the Artin generators $\sigma_1,\dots,\sigma_{n−1}$. Furthermore, the kernel of the surjection from the free $B_n$-crossed module $C_n$ in any one of the $\sigma_j$’s onto $B_n$ coincides with the second homology group $H_2(B_n)$ of $B_n$, well known to be cyclic of order $2$ when $n\ge 4$ and trivial for $n = 2$ and $n = 3$.

Answer 3

I can't resist referring to the paper

R. Brown and P.J. Higgins, ``On the connection between the second relative homotopy groups of some related spaces'', {\em Proc. London Math. Soc.} (3) 36 (1978) 193-212,

which gives some explicit calculations with pushouts of crossed modules, but which is little referred to, and further examples are in

(with C.D.WENSLEY), `Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72.

The key to the proof of the theorem is the notion of homotopy double groupoid of a based pair.

The paper arXiv:0909.3387v2 gives applications of higher van Kampen theorems to homotopy groups of spheres.

January 22, 2016 Tom refers to "an extra bit of structure". This structure is crucial for applications of a colimit theorem (see the above 1978 paper, and the 2011 book partially titled 2011 book Nonabelian Algebraic Topology, NAT, pdf available); altering the structure alters the colimits. Part I of this book gives lots of explicit calculations, for example showing how for $A$ connected, and $f:A \to X$ a pointed map, the crossed module $$\pi_2(X \cup _f CA,X,x) \to \pi_1(X,x)$$is determined by the morphism $f_*:\pi_1(A) \to \pi_1(X)$. This is a generalisation of a theorem of JHC Whitehead in "Combinatorial Homotopy II" on free crossed modules, which is the case $A$ is a wedge of circles.

There is a general assumption that we most want to calculate the second homotopy group $\pi_2$; but this, even as a module over $\pi_1$ is but a pale shadow of the 2-type.

One of the problems in homotopy theory is that identifications in low dimensions have influence on high dimensional invariants. So part of the aim is to study this influence using algebraic structures which have information in a range of dimensions, and to develop and apply nonabelian colimit theorems in higher homotopy.

Tim Porter refers to the nonabelian tensor square, which generalises to tensor products. I have kept up a bibliography on this topic; the idea arose from considering pushouts of crossed squares, and the bibliography currently has 144 items, dating back to 1952.

January 23,2016 I thought of another point which maybe answers the question better, namely as to the import of these crossed module rules on this boundary map for second relative homotopy groups. The situation becomes clearer if you work with the homotopy double groupoid of a pointed pair of spaces, see the NAT book and also the presentation at Galway on my preprint page. In terms of double groupoids, the first rule is a boundary rule for a certain subdivided square, and the second rule is equivalent to the interchange law for the two compositions. So if you draw the right pictures, then these rules become necessities.

The emphasis on homotopy groups is part of the "squashing" of 2-dimensional situation into a single dimension, on a line. Then some natural rules become obscure, and it becomes impossible to do the 2-dimensional compositions which are necessary for the proof of the Seifert-van Kampen Theorem for homotopy double groupoids and so for the homotopy crossed module of the question.