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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 $\sigma$-compact space $X$, and with a base ray $* : [0,\infty) \to X$, with end $\varepsilon(X)$, the Čech homotopy groups of $\varepsilon(X)$ relative to $*$ are defined by $lim \pi_n(\varepsilon(X))$. 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:
Let $\underline{S}^n = ([0,\infty) \cup \bigcup_{k=0}^\infty(S^n \times \{k\})/(k\sim (\underline{1},k)$. This is called a string of $n$-spheres.
For the string of circles, it looks something like this:
$\varepsilon(\underline{S}^n)$ is a pro-space isomorphic to an $\mathbb{N}$-indexed one with $k^{th}$ level, $\bigvee_{j\geq k}S^n_j$, where each $S^n_j$ is a copy of $S^n$. The structural/ bonding maps from the $(k+1)^{st}$ level to the $k^{th}$ level are the obvious inclusions.
The space $\underline{S}^n$ is, of course, considered as being based at the ray (given by the right vertical map in the pushout square).
The space $\underline{S}^n$ is a spherical object and the family $\mathcal{A} = \{\underline{S}^n\}^\infty_{n=0}$ defines a theory.
The $n^{th}$ Brown-Grossman homotopy group of $(X,*)$ is given by the group of based proper homotopy classes of based proper maps from $\underline{S}^n$ to $(X,*)$;
As was pointed out above, the $\underline{S}^n$ are cogroups so the $\underline{\pi}_n(X,*)$ are groups, which are abelian for $n\geq 2$.
In fact the $\underline{\pi}_n(X,*)$ form a $\Pi_\mathcal{A}$-algebra for $\mathcal{A}$ as above. Not only are there the sort of morphisms in $\Pi_\mathcal{A}$ that will induce analogues of Whitehead products, composition operations etc. but there are other interesting morphisms there, for instance:
There is a proper shift map $S: \underline{S}^n\to \underline{S}^n$, which shifts all the spheres one place to the right. This induces, by composition, a morphism $S^*(X): \underline{\pi}_n(X,*)\to \underline{\pi}_n(X,*)$. This means that each individual $\underline{\pi}_n(X,*)$ has more structure than simply being a group.
The $n^{th}$ Brown-Grossman homotopy group of $(X,*)$ at $\infty$ is given by the group of based germ homotopy classes of based proper germs from $\underline{S}^n$ to $(X,*)$;
With this description, it is clear that $\underline{\pi}^\infty_n$ is functorial on base rayed spaces and that it only depends on the choice of the class of $*$ within $e(X)$ (cf. proper homotopy theory).
There are several different types of generalisation of Whitehead’s theorem to the proper homotopy setting. The following is due to Ed Brown (1974):
(Brown, 1974) Let $K$, $L$ be finite dimensional connected locally finite simplicial complexes, then a proper map $f : K \to L$ is a proper homotopy equivalence if, and only if:
$e(f) : e(K)\to e(L)$ is a homeomorphism;
for each $n$ $\pi_n(f) : \pi_n(K,*(0))\to \pi_n(L,f(*(0)))$ is an isomorphism;
for each $n$, and for each base ray, $*$, in $K$, $\underline{\pi}^\infty_n(K,*)\to \underline{\pi}^\infty_n(L,f(*))$ 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).)
It would be useful to have a construction of the groups $\underline{\pi}^\infty_n(X,*)$ from the pro-group $\pi_n(\varepsilon(X),*(k))$. 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 $\underline{G} = \{G_n,p^m_n\}$ 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 $G_0=1$. Consider all sequences $\{g_{k(n)}\}$ with $g_{k(n)} \in G_{k(n)}$, where $k(n)$ is a sequence of natural numbers such that $k(n)\to \infty$ as $n\to \infty$. Given two such sequences $\{g_{k(n)}\}$ and $\{g\prime_{l(n)}\}$, we say they are equivalent if there is a third sequence $m(n)$, $m(n)\to \infty$ as $n \to \infty$, with $m(n)\leq min(k(n),l(n)))$ and $p^{k(n)}_{m(n)}(g_{k(n)}) = p^{l(n)}_{m(n)}(g\prime_{l(n)})$. We let $\mathcal{P}(\underline{G})$ be the set of equivalence classes.
This has a natural group structure;
If $X = \bigcup_n K_n$, $U_n = X- K_n$, and $* : [0,\infty) \to X$ is chosen so that $*[n,\infty)\subset U_n$, then setting $G_n = \pi_k(U_n,*(n))$ with $G_n\to G_{n-1}$ induced by the inclusion of $U_n$ into $U_{n-1}$ and the change of base point along $*([n-1,n])$, then $\underline{\pi}^\infty_n(X,*) \cong \mathcal{P}(\underline{G}).$
For any tower of groups, $\underline{G}$ there is an action of the group $F = \underline{\pi}_1(\underline{S}^1,[0,\infty)$ on $\mathcal{P}(\underline{G})$.
(Chipman) Let $\underline{G}$, $\underline{H}$ be towers of finitely generated groups, then $\underline{G}$ is isomorphic to $\underline{H}$ if, and only if, there is an isomorphism from $\mathcal{P}(\underline{G})$ to $\mathcal{P}(\underline{H})$ commuting with the operation of $F$. (What is remarkable here is that initially no morphism between $\underline{G}$ and $\underline{H}$ is given. It is constructed from the ones on the images under $\mathcal{P}$.)
Grossman gave an alternative construction of $\mathcal{P}(\underline{G})$ for any tower of groups (or, in fact, any tower of sets, $\underline{X}$). This involves the reduced product.
General references include: the survey article:
and for a slightly different approach:
A specific reference for the Brown Whitehead theorem is
and further