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Brown-Grossman homotopy group

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Idea: use a string of spheres not just one!

Within the context of proper homotopy theory, what might be the analogues of the classical homotopy groups? We might hope that Čech style homotopy groups would work, but they don’t, or rather, ‘not exactly’!

Given a locally compact and σ\sigma-compact space XX, and with a base ray *:[0,)X* : [0,\infty) \to X, with end ε(X)\varepsilon(X), the Čech homotopy groups of ε(X)\varepsilon(X) relative to ** are defined by limπ n(ε(X))lim \pi_n(\varepsilon(X)). These ‘make sense’ but do not behave well, since the limit will destroy exactness in situations where we would hope for, and expect, long exact sequences. The problem is related to the fact that, depending how you view this, we are using just one sphere, or alternatively an infinite sequence of unrelated spheres.

E.M. Brown (1974) suggested a different construction based on an infinite string of spheres and this helps reveal extra structure that helps:

Strings of spheres

Definition

Let S̲ n=([0,) k=0 (S n×{k})/(k(1̲,k)\underline{S}^n = ([0,\infty) \cup \bigcup_{k=0}^\infty(S^n \times \{k\})/(k\sim (\underline{1},k). This is called a string of nn-spheres.

This, of course, means that S̲ n\underline{S}^n is defined as a pushout

For the string of circles, it looks something like this:

Properties

  1. ε(S̲ n)\varepsilon(\underline{S}^n) is a pro-space isomorphic to an \mathbb{N}-indexed one with k thk^{th} level, jkS j n\bigvee_{j\geq k}S^n_j, where each S j nS^n_j is a copy of S nS^n. The structural/ bonding maps from the (k+1) st(k+1)^{st} level to the k^{th level are the obvious inclusions.

  2. The space S̲ n\underline{S}^n is, of course, considered as being based at the ray (given by the right vertical map in the pushout square).

  3. The space S̲ n\underline{S}^n is a spherical object and the family 𝒜={S̲ n} n=0 \mathcal{A} = \{\underline{S}^n\}^\infty_{n=0} defines a theory.

The Brown-Grossman homotopy groups

Definition

The n thn^{th} Brown-Grossman homotopy group of (X,*)(X,*) is given by the group of based proper homotopy classes of based proper maps from S̲ n\underline{S}^n to (X,*)(X,*);

π̲ n(X,*)=Ho(Proper)(S̲ n,(X,*)).\underline{\pi}_n(X,*) = Ho(Proper)(\underline{S}^n, (X,*)).

Properties

  1. As was pointed out above, the S̲ n\underline{S}^n are cogroups so the π̲ n(X,*)\underline{\pi}_n(X,*) are groups, which are abelian for n2n\geq 2.

  2. In fact the π̲ n(X,*)\underline{\pi}_n(X,*) form a Π 𝒜\Pi_\mathcal{A}-algebra for 𝒜\mathcal{A} as above. Not only are there the sort of morphisms in Π 𝒜\Pi_\mathcal{A} that will induce analogues of Whitehead products, composition operations etc. but there are other interesting morphisms there, for instance:

  3. There is a proper shift map S:S̲ nS̲ nS: \underline{S}^n\to \underline{S}^n, which shifts all the spheres one place to the right. This induces, by composition, a morphism S *(X):π̲ n(X,*)π̲ n(X,*)S^*(X): \underline{\pi}_n(X,*)\to \underline{\pi}_n(X,*). This means that each individual π̲ n(X,*)\underline{\pi}_n(X,*) has more structure than simply being a group.

The variant ‘at the end’

The Brown-Grossman homotopy groups at \infty

Definition

The n thn^{th} Brown-Grossman homotopy group of (X,*)(X,*) at \infty is given by the group of based germ homotopy classes of based proper germs from S̲ n\underline{S}^n to (X,*)(X,*);

π̲ n (X,*)=Ho(Proper )(S̲ n,(X,*)).\underline{\pi}^\infty_n(X,*) = Ho(Proper_\infty)(\underline{S}^n, (X,*)).

With this description, it is clear that π̲ n \underline{\pi}^\infty_n is functorial on base rayed spaces and that it only depends on the choice of the class of ** within e(X)e(X) (cf. proper homotopy theory).

A proper Whitehead theorem

There are several different types of generalisation of Whitehead’s theorem to the proper homotopy setting. The following is due to Ed Brown (1974):

Theorem

(Brown, 1974) Let KK, LL be finite dimensional connected locally finite simplicial complexes, then a proper map f:KLf : K \to L is a proper homotopy equivalence if, and only if:

  1. e(f):e(K)e(L)e(f) : e(K)\to e(L) is a homeomorphism;

  2. for each nn π n(f):π n(K,*(0))π n(L,f(*(0)))\pi_n(f) : \pi_n(K,*(0))\to \pi_n(L,f(*(0))) is an isomorphism;

  3. for each nn, and for each base ray, **, in KK, π̲ n (K,*)π̲ n (L,f(*))\underline{\pi}^\infty_n(K,*)\to \underline{\pi}^\infty_n(L,f(*)) is an isomorphism.

If one removes the condition of finite dimensionality, the result no longer holds. (There is an error in Brown’s subsequent reasoning in the quoted paper, for which one needs to consult Edwards and Hastings, (1976).)

Brown’s 𝒫\mathcal{P}-functor

It would be useful to have a construction of the groups π̲ n (X,*)\underline{\pi}^\infty_n(X,*) from the pro-group π n(ε(X),*(k))\pi_n(\varepsilon(X),*(k)). Such a construction was given by Brown in the same article (1974). (An alternative construction due to Grossman will be discussed in a separate entry.)

Let G̲={G n,p n m}\underline{G} = \{G_n,p^m_n\} be an inverse sequence of groups (aka tower of groups), that is a pro-group that is indexed by the ordered set of positive integers. We assume G 0=1G_0=1. Consider all sequences {g k(n)}\{g_{k(n)}\} with g k(n)G k(n)g_{k(n)} \in G_{k(n)}, where k(n)k(n) is a sequence of natural numbers such that k(n)k(n)\to \infty as nn\to \infty. Given two such sequences {g k(n)}\{g_{k(n)}\} and {g l(n)}\{g\prime_{l(n)}\}, we say they are equivalent if there is a third sequence m(n)m(n), m(n)m(n)\to \infty as nn \to \infty, with m(n)min(k(n),l(n)))m(n)\leq min(k(n),l(n))) and p m(n) k(n)(g k(n))=p m(n) l(n)(g l(n))p^{k(n)}_{m(n)}(g_{k(n)}) = p^{l(n)}_{m(n)}(g\prime_{l(n)}). We let mathcalP(G̲)\,mathcal{P}(\underline{G}) be the set of equivalence classes.

  • This has a natural group structure;

  • If X= nK nX = \bigcup_n K_n, U n=XK nU_n = X- K_n, and *:[0,)X* : [0,\infty) \to X is chosen so that *[n,)U n*[n,\infty)\subset U_n, then setting G n=π k(U n,*(n))G_n = \pi_k(U_n,*(n)) with G nG n1G_n\to G_{n-1} induced by the inclusion of U nU_n into U n1U_{n-1} and the change of base point along *([n1,n])*([n-1,n]), then π̲ n (X,*)𝒫(G̲).\underline{\pi}^\infty_n(X,*) \cong \mathcal{P}(\underline{G}).

  • For any tower of groups, G̲\underline{G} there is an action of the group F=π̲ 1(S̲ 1,[0,)F = \underline{\pi}_1(\underline{S}^1,[0,\infty) on 𝒫(G̲)\mathcal{P}(\underline{G}).

  • (Chipman) Let G̲\underline{G}, H̲\underline{H} be towers of finitely generated groups, then G̲\underline{G} is isomorphic to H̲\underline{H} if, and only if, there is an isomorphism from 𝒫(G̲)\mathcal{P}(\underline{G}) to 𝒫(H̲)\mathcal{P}(\underline{H}) commuting with the operation of FF. (What is remarkable here is that initially no morphism between G̲\underline{G} and H̲\underline{H} is given. It is constructed from the ones on the images under 𝒫\mathcal{P}.)

Grossman’s reduced product construction

Grossman gave an alternative construction of 𝒫(G̲)\mathcal{P}(\underline{G}) for any tower of groups (or, in fact, any tower of sets, X̲\underline{X}). This involves the reduced product?.

References

General references include: the survey article:

  • Tim Porter, Proper Homotopy Theory, in the Handbook on Algebraic Topology, Ed. I.M.James, Elsevier, 1995, p. 127-167,

and for a slightly different approach:

  • Hans Baues and Antonio Quintero, Infinite homotopy theory, Volume 6 of K-monographs in mathematics, Springer, 2001

A specific reference for the Brown Whitehead theorem is

  • E. M. Brown, Proper homotopy theory in simplicial complexes, Topology Conference (Virginia Polytechnic Institute and State University, Dickmann and Fletcher, eds.) SLNM 375, Springer (1974), pp. 41-46,

and further

  • D. A. Edwards and H. M. Hastings, Čech and Steenrod Homotopy Theory with Applications, SLNM 542, Springer (1976).

Revised on November 15, 2010 18:56:38 by Tim Porter (95.147.238.71)