CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Proper homotopy theory is both an old and a fairly new area of algebraic topology. It deals with properties of non-compact topological spaces, that cannot be detected using maps from simplices in these spaces, such as used in the singular complex.
Historically the subject is traced back to Kerekjarto’s classification of non-compact surfaces in 1923, but its emergence as an important tool in geometric topology came with Larry Siebenmann’s work in 1965.
Suppose $M$ is a smooth manifold with boundary, $\partial M$, then $M\setminus \partial M$ is an open manifold. Now suppose someone gives us an open manifold $N$, is it possible to detect if there is a compact manifold $M$, with $M\setminus \partial \cong N$. Siebenmann showed that certain conditions on the ends of $N$ were necessary and that there were obstructions if the dimension of $M$ was greater than 5.
Its potential importance of proper homotopy, for example for physical applications, comes from the fact that the phenomena it studies include the limiting behaviour of the system.
The basic hypothesis will be that $X$ will be a connected and locally connected compact Hausdorff space. It will usually be $\sigma$-compact, i.e., there will be an increasing sequence, $\{K_n\}$, of compact subspaces with each $K_n$ in the interior of $K_{n+1}$ and such that
These spaces will most often than not be locally finite simplicial complexes.
We will be interested in the homotopy of such spaces ‘out towards its ends’
To illustrate the idea of the ends of a space $X$, we note that naively $\mathbb{R}$ has two ends, $\infty$ and $-\infty$, whilst $\mathbb{R}^2$ has only one as it is $S^2\setminus \{\infty\}$, (but that is vague!).
More exactly, consider the system of spaces
This is an inverse system or pro-object in the category of spaces. Applying the connected component functor, $\pi_0$, to this system of spaces gives $\pi_0(X)$, and, classically, one takes the limit of this to get
the set of ends of $X$. In general, $e(X)$ would be given the inverse limit topology?, to preserve more of the information coming from its construction. This space is the space of (Freudenthal) ends of $X$. It is a profinite space.
Let $X_8$ be the figure eight space, the one-point union of two circles, and let $X$ be its universal cover. This is an infinite ‘thorn bush’. It has infinitely many ends and
Let $M$ be a compact manifold and $X = M\setminus \partial M$, then $e(X) \cong \pi_0(\partial M)$.
The assignment sending $X$ to $e(X)$ cannot be functorial on the category of spaces and continuous maps, since the contracting map $\mathbb{R}\to \{0\}$ is continuous, $e(\mathbb{R})$ is $\{-\infty, \infty\}$, whilst $e(\{0\})$ is empty since $\{0\}$ is compact. The problem is that continuity is really about inverse images (inverse image of open is open), but the inverse image mapping does not preserve compactness (as in the example!).
A map $f:X\to Y$ is a proper map if for each subset $K$ compact in $Y$ then $f^{-1}(K)$ is compact in $X$.
If $f:X\to Y$ is proper, then it induces a pro-morphism
and hence a continuous map of the end spaces
and $e$ becomes a functor from some category $Proper$ of spaces and proper maps to $Stone$, the category of profinite spaces / Stone spaces.
For any space $X$, the natural inclusions of $X$ into $X \times I$, $e_i(x) = (x,i)$, $i = 0,1$ together with the projection maps from $X\times I$ to $X$, are proper maps.
The category $Proper$ has a cylinder functor.
We call the corresponding notion of homotopy, ‘proper homotopy’. We get a ‘Proper category’ and an associated ‘proper homotopy category’, which we will denote $Ho(Proper)$. If we are restricting to $\sigma$-compact spaces we may write $Proper_\sigma$, and so on.
Although a proper map $f: X\to Y$ will induce a continuous $\varepsilon(f)$, on the end spaces, it is clear that $f$ does not need to be defined on the whole of $X$ for this to work, as $\varepsilon$ encodes behaviour ‘out towards $\infty$’. This leads to the notion of a ‘germ at $\infty$’.
Suppose $X$ is locally compact Hausdorff and $A\subset X$. The inclusion $j: A\to X$ is ‘cofinal’ if the closure of $X\setminus A$ is compact. Note that a cofinal inclusion is proper and induces an isomorphism $\varepsilon(A)\cong \varepsilon(X)$. Let $\Sigma$ be the class of all cofinal inclusions in $Proper$ and let $Proper_\infty = Proper[\Sigma^{-1}]$, the category obtained by formally inverting the cofinal inclusions.
This category is called the proper category at $\infty$.
Note that $(Proper,\Sigma)$ admits a calculus of right fractions, so any morphism from $X$ to $Y$ in $Proper_\infty$ can be represented by a diagram
with $j$ a cofinal inclusion, i.e., $f$ is defined on some ‘neighbourhood of the end of $X$’.
Two such diagrams
represent the same germ if $f^\prime | A = f^{\prime\prime}|A$ for some cofinal subspace $A$ with $A^\prime \cup A^{\prime\prime}\subset A$.
There is also a homotopy category $Ho(Proper_\infty)$
The end space $e(X)$ is a Stone space so is $Max(R)$ for some Boolean algebra $R$ (Stone duality) In the 1960s someone (Goldman?) looked at a ring, $R$, of ‘almost continuous functions’ from $X$ to $\mathbb{Z}/2\mathbb{Z}$, that gave the right $e(X)$. Can this idea help integrate better the ideas of proper homotopy etc. with modern methods of algebraic geometry?
The end space behaves a bit like a $\pi_0$ and usually spaces will have many ends, so are not ‘connected at infinity’. If we try to do a fundamental group or groupoid analogue, this means life will get more complicated. We will try with the assumption of a space having a single end for simplicity! We will also assume $X$ is $\sigma$-compact.
We could try defining $\pi_1(\varepsilon(X))$ as a progroup, then taking its limit. For this we would take $\{K_n\}$ an exhausting increasing sequence of compact subsets and setting $U_i = X\setminus K_i$, pick a base point $x_i$ in each $U_i$, and we will get groups $\pi_1(U_i,x_i)$. We however need induced homomorphisms $\pi_1(U_{i+1},x_{i+1})\to \pi_1(U_i,x_i)$, and for this we have to choose an arc in $U_i$ from $x_{i+1}$ to $x_i$. We can combine these to get a base ray, rather than a base point, that is, we need a proper map, $\alpha : [0,\infty)\to X$. With that we do get an inverse sequence of groups, but there are problems. What is the dependence of the inverse system on the choice of $\alpha$?
Let $X$ be an infinite cylinder with an infinite string of circles attached via a proper ray $\alpha: [0,\infty) \to X$. The space has just one ‘end’ but you can choose different ways of going from $\pi_1(U_{i+1},x_{i+1})$ to $\pi_1(U_i,x_i)$ for fairly obvious choices of base points such that the limit groups of the resulting two inverse systems are non-isomorphic! (In the survey listed below, this example is examined in detail, and one of the limits is a free group on one element, the other is trivial! Definitely non-isomorphic!)
This means that $lim \pi_1(\varepsilon(X))$ is not an invariant of the end. This phenomenon is linked to the fact that $\pi_1(\varepsilon(X))$ does not satisfy the Mittag-Leffler condition for either choice of the base rays.
If $X$ and $Y$ are locally compact Hausdorff spaces, there is no obvious candidate for a space of proper maps from $X$ to $Y$, but one can form a simplicial set $\mathbb{P}(X,Y)$ with $\mathbb{P}(X,Y)_n = Proper(X\times \Delta^n,Y)$, which acts as if it was the singular complex of the mythical space of proper maps from $X$ to $Y$.
The Waldhausen boundary of $X$ is the simplicial set $\mathbb{P}([0,\infty),X)$.
There is an epimorphism from $\pi_0(\mathbb{P}([0,\infty),X))$ to $e(X)$.
In the example above of the cylinder with the string of circles attached, $\pi_0(\mathbb{P}([0,\infty),X))$, is uncountable, and $\pi_1(\mathbb{P}([0,\infty),X))$ maps onto $lim S$.
When $X$ has a single end and $\pi_0(\varepsilon(X))$ is Mittag-Leffler, then $\pi_0(\mathbb{P}([0,\infty),X))$, is a single point, i.e. all possible base rays are properly homotopic.
Even if we did not have the above difficulty with the limit groups, we would still have the problem that, as the limit functor is not exact, the resulting limiting homotopy groups would not be that well behaved. There would not be any general long exact sequence results (just as with Čech homology). There is at least one possible replacement for those limiting homotopy groups, but first we note that it is not appropriate to base any such things at a point, rather we should be using a base ray as was discussed above.
One fairly obvious attempt to define a ‘fundamental group’ for $X$, based at a proper ray $\alpha: [0,\infty) \to X$, would be to note that $\alpha$ gives $\mathbb{P}([0,\infty),X)$ a base point so we could look at $\pi_1(\mathbb{P}([0,\infty),X),\alpha)$ and more generally at $\pi_n(\mathbb{P}([0,\infty),X),\alpha)$, and we will denote these groups by $\underline{\underline{\pi}}_n(X,\alpha)$.
There are variants ‘at infinity’ of both the Waldhausen boundary and these groups, otained using germs instead of proper maps. These will be denoted with a $\infty$ as a super- or suffix on the above notation.
(Once over lightly here, more details at Brown-Grossman homotopy group.) In fact there is another different way of looking at these groups, which has a more geometric feel to it. Historically these groups were not the first successful attempt. This was due to Ed Brown and uses strings of spheres. These are examples of spherical objects and the resulting ‘groups’ (better thought of as ‘$\Pi_\mathcal{A}$-algebras’) have a rich structure. They are discussed at Brown-Grossman homotopy groups. They do link with the above homotopy groups of the Waldhausen boundary, which are called the Steenrod homotopy group?s, (for reasons that will be explained there).
Survey article:
Books:
H. J. Baues and A. Quintero, Infinite homotopy theory, Volume 6 of K-monographs in mathematics, Springer, 2001
Bruce Hughes and Andrew Ranicki, Ends of Complexes, Cambridge Tracts in Mathematics (No. 123), C.U.P.
Lecture Notes: