Homotopy classes of proper maps out of vector bundles

In this paper we classify the homotopy classes of proper maps $E\rightarrow \mathbb R^k$, where $E$ is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps $\mathbb R^n\rightarrow \mathbb R^k$. We find a stability range of such maps. We conclude with some remarks on framed submanifolds of non-compact manifolds, the relationship with proper homotopy classes of maps and the Pontryagin-Thom construction.


Introduction.
A continuous map f : X → Y is called proper if f −1 (C) is compact for all compact subsets C of Y . A homotopy of proper maps is a homotopy F : [0, 1] × X → Y such that F is a proper map. The assumption that a homotopy is a homotopy of proper maps is stronger than the assumption that the homotopy is homotopy through proper maps, i.e. the assumption that the maps F t : X → Y are proper for every t ∈ [0, 1]. A simple example of a homotopy through proper maps that is not a homotopy of proper maps is the map F : [0, 1] × R → R defined by F (t, x) = tx 2 + x. To see this, note that the sequence ( 1 n , −n) is unbounded, but F ( 1 n , −n) = 0. This example is closely related to the compactness issues discussed in [5].
We denote by [X, Y ] the set of (unbased) homotopy classes of maps from X to Y and with [X, Y ] prop the set of (unbased) homotopy classes of proper maps. For the set of homotopy classes of based maps between pointed spaces we write X, Y .
In [1], we classified the homotopy classes of proper Fredholm maps of Hilbert manifolds into its model (real and separable) Hilbert space in terms of a suitable notion of framed cobordism. This classification uses an infinite dimensional and proper analogue of the Pontryagin-Thom collapse map, which is due to Elworthy and Tromba [3], see also the paper of Gęba [4]. The existence of the collapse map hinges on the fact that an infinite dimensional Hilbert space is diffeomorphic to the Hilbert space minus a point. This is of course not true for a finite dimensional vector space and the construction does not work in this setting. As we will discuss in Section 4, even though the framed cobordism class of a regular value is an invariant of the homotopy class of a proper map in the finite dimensional setting, the framed cobordism class is not able to distinguish all proper homotopy classes of proper maps into R k , nor do all framed submanifolds come from proper maps.
We have the following corollary of Theorem 1.1 by taking M to be a point and using the fact that based and unbased homotopy classes of maps from spaces to positive dimensional spheres coincide, cf. [6, Section 4A].
Thus if n > 1 and k > 1, we have that [R n , R k ] prop is in bijection with π n−1 (S k−1 ). The set [R n , R] prop has two elements if n > 1 and four elements if n = 1.
A proper map between non-compact and locally compact Hausdorff spaces extends to a continuous map between the one point compactifications by sending infinity to infinity. Similarly a homotopy of proper maps induces a homotopy in the one point compactification.
The one point compactification of a real vector bundle E → M over a compact Hausdorff space M equals the Thom space Th(E) of the vector bundle and the one point compactification of R k is homeomorphic to S k by stereographic projection. Thus we obtain a map Q : In Section 3, we show that the map Q is bijective if k is sufficiently large. If E = R n , the map Q is nothing but the suspension π n−1 (S k−1 ) → π n (S k ) under the identification of [R n , R k ] prop and π n−1 (S k−1 ) of Corollary 1.2.
For l sufficiently large, the sets [E ⊕R l , R k+l ] prop and [E ⊕R l+1 , R k+l+1 ] prop are in bijection. Thus it makes sense to define the stable proper homotopy classes as which are in bijection with the stable cohomotopy groups π k S (Th(E)), cf. Corollary 3.2. Using Atiyah duality, we obtain the following result.
We write S(E) for S 1 (E) and B(E) for B 1 (E).
Given a homotopy F : Compact subsets of E are characterized as follows: A subset K ⊆ E is compact if and only if it is closed and bounded. Here bounded means that K ⊆ B R (E) for some R > 0. As P F (·, t) is norm preserving for all t, it follows that P F is proper. The same construction assigns to a map f : We will show that P is bijective. Let us start with the injectivity. We need to show that f 0 and f 1 are homotopic if g 0 = P f 0 and g 1 = P f 1 are homotopic as proper maps. Let G : [0, 1] × E → R k be a homotopy of proper maps between g 0 and g 1 . Then there exists an R > 0 such that is a homotopy between f 0 and f 1 , hence P is injective.
To show that P is surjective, we need to show that, given a proper map . As h is homotopic as proper map to the identity via (t, x) → (1 − t)h(x) + x, it follows that the map g 1 is proper homotopic to g. Note that if ||v|| < 1, then ||Rv|| < R and therefore ||g(Rv)|| < 1. We see that g 1 (B(E)) ⊆ B(R k ) and similarly that For this, it is sufficient to show that for all s > 1, there exists an S > 1 such that

Suppose the set of solutions of this equation is not contained in
for any S. Then we have a sequence (t n , v n ) of solutions such that ||v n || ≥ n. Without loss of generality we take a subsequence such that t n converges to t by the compactness of [0, 1]. This subsequence will also satisfy ||v n || ≥ n. If t > 0, then there exists an N such that for all n ≥ N , we have t n > t 2 and which contradicts the unboundedness of v n . If t = 0, then there exists an N such that for all n ≥ N , the sequence satisfies t n < 1 2 and Vol. 114 (2020) Homotopy classes of proper maps out of vector bundles 111 The sequence g 1 (v n ) is therefore bounded and as the map g 1 is proper, it follows that the sequence v n is also bounded. This contradicts the assumption that v n is unbounded. This means that G 2 [0,1]×(E\B(E)) is proper. Thus P f is proper homotopic to g and P : We have already shown that P is injective and Theorem 1.1 follows.

The one
In the proof of Theorem 1.1, we saw that the maps P are bijections. We now investigate when the other maps in the diagram are bijective.
Let us now assume that k ≥ 2, that M is a finite connected CW-complex of dimension m, and that E is a normed real vector bundle of rank n. Since k ≥ 2, based and unbased homotopy classes into S k−1 and S k coincide, as well Arch. Math.
as based or unbased proper homotopy classes into R k and R k+1 . We denote by π k−1 (S(E)) := S(E), S k−1 the (k − 1)-th cohomotopy set of S(E). We refer to [7, Chapter VII] for information on the cohomotopy sets we use below. The cohomotopy set π k−1 (S(E)) is not always a group, but it is if m + n ≤ 2k − 3. We investigate the long exact sequence of the pair (B(E), S(E)) if m + n ≤ 2k − 3: Since B(E) deformation retracts to M and S k−1 is (k − 2)-connected, we see that if m ≤ k − 2, there are isomorphisms Thus we conclude that for 2k ≥ m + 3 + max(n, m + 1), there is an isomorphism π k−1 (S(E)) ∼ = π k (B(E), S(E)). The relative cohomotopy set is the cohomotopy set of the quotient for nice spaces, thus π k (B(E), S(E)) = π k (B(E)/S(E)) = π k (Th(E)). The coboundary map is an isomorphism π k−1 (S(E)) ∼ = π k (Th(E)) in the dimension range. Let us consider the based version of Diagram (1) The horizontal maps can be identified with the coboundary map δ in (2) and therefore the horizontal maps are isomorphisms in the right dimension range. Freudenthal's suspension theorem, cf. [8], states that if m + n ≤ 2k − 2, the suspension map π k (Th(E)) → π k+1 (S Th(E)) ∼ = π k+1 (Th(E ⊕ R)) is an isomorphism. Combining all this information gives us the following theorem. This theorem expresses a stability phenomenon: For all l sufficiently large, We define the stable homotopy classes of proper maps as Recall that the stable homotopy and cohomotopy groups of a space X are similarly defined π S k (X) = lim l→∞ S l S k , S l X and π k S (X) = lim l→∞ S l X, S l S k .
A direct corollary of Theorem 3.1 is then  (Th(E)). Stable homotopy and cohomotopy groups are related via Spanier-Whitehead duality, which we recall now. We refer to the original references [11,12] for these statements. Let i : X → S N be a sufficiently nice embedding of a sufficiently nice space X into a sphere (e.g. a smooth embedding of a compact manifold, or the inclusion of a CW complex as a subcomplex). Then the space D N X = S N \ i(X) is a Spanier-Whitehead dual of X. The stable homotopy type of D N X is well defined: It is independent of the dimension N and the choice of embedding. The fundamental result is that lim In particular, the stable cohomotopy groups of X are the stable homotopy groups of D N X with a dimension shift. Now let us assume that M is a compact manifold with boundary ∂M . There is a unique (up to isotopy) embedding of M into R m+n for n sufficiently large. Let E be the normal bundle of such an embedding, i.e. let E be the stable normal bundle of M . Atiyah [2,Proposition 3.2] showed that

Framed submanifolds and cobordisms.
Pontryagin [10] showed that homotopy classes of maps M → S k , where M is a closed manifold, are in one to one correspondence with framed cobordism classes of (n − k)-dimensional manifolds in M . Framed cobordism classes are also invariants of homotopy classes of proper maps E → R k but they are not complete, nor is every cobordism classed realized by some proper map. In this section, we discuss this.  x(2+sin(x)) , but of course this map fails to be proper and closed. The preimage of a regular value y is a closed submanifold X = f −1 (y) of dimension m − k. Such a manifold can be framed : Let e 1 , . . . , e k be a basis of T y N that is compatible with the orientation of N . Then for every x ∈ X, the differential of f induces an isomorphism df x : is an ordered basis of the normal space N x X at x. Letting x vary, this patches together to a map ν f that trivializes the normal bundle of X. The map ν f is called the framing of X. We call (X, ν f ) a Pontryagin manifold of f and it depends on the choices we made.
Let F : [0, 1] × M → N be a homotopy of proper maps between f 0 = F (0, ·) and f 1 = F 1 (1, ·). By a reparametrization of the homotopy variable, we may assume that F (t, x) = f 0 (x) and F (1 − t, x) = f 1 (x) for t small. If y is a regular value of the maps F, f 0 , and f 1 simultaneously, then (W = F −1 (y), ν F ) is a framed compact submanifold with framed boundary (X 0 = f −1 0 (y), ν f0 ) and (X 1 = f −1 0 (y), ν f1 ). The framed manifold (W, ν F ) is a framed cobordism between the framed manifolds (X 0 , ν f0 ) and (X 1 , ν f1 ). Being framed cobordant, defines an equivalence relation on the set of framed submanifolds and the framed cobordism class of a Pontryagin manifold of a proper map f : M → N does not depend on the choice of the regular value y and the choice of the oriented basis of T y N and is an invariant of the proper homotopy class of f . We denote the set of framed (m−k)-dimensional closed submanifolds of M up to framed cobordism by Ω fr m−k (M ).

The Pontryagin-Thom construction.
The framed cobordism class of the preimage of a regular value is in some cases enough to recover the homotopy class of the map: Suppose M is closed and (X, ν) is a (m − k)-dimensional framed submanifold. Out of this data we can construct a (proper) map f : M → S k , for which (X, ν) is a Pontryagin manifold: We define f to map X to the northpole y of S k and describe what happens in a tubular neighborhood of X. The framing ν defines, for each point x ∈ X, a diffeomorphism of the normal space around x to a neighborhood of y. We use this to extend the map to the tubular neighborhood of X in M . One can arrange this in such a way that if one approaches the boundary of the tubular neighborhood, the image under f converges to the south pole. The map f can now be extended to the whole of M by mapping everything outside the tubular neighborhood to the south pole. The northpole is a regular value for f and the Pontryagin manifold at the north pole is exactly the framed manifold (X, ν). This construction also works for framed cobordisms. This proves the following theorem.