INDUCED FIBRATIONS ON SPACES OF FIBER TRANSFERRING MAPS: THE C^{r} CASE

Manolis Magiropoulos

Abstract. For C^{r} differentiable manifolds E and B of the same dimension, and π: E → B a local C^{r} diffeomorphism and onto, we examine, as with the topological case, the extent to which some basic properties of π are inherited by the induced map p on spaces of fiber transferring maps with respect to π.

If E and B are two connected, C^{r} differentiable manifolds (r = 1, 2, …, ∞) of the same dimension, and π: E → B is a local C^{r} diffeomorphism and onto, then one can consider the C^{r} counterparts of the spaces C_{π}(E), H_{π}(E), Top_{π}(E), C(B), H(B) and Top(B) considered in [2]. The new spaces are denoted by C_{π}^{r}(E), H_{π}^{r}(E), Diff_{π}^{r}(E), C^{r}(B), H^{r}(B), and Diff^{r}(B), respectively. They are all given the weak or the strong C^{r} topology [1]. The two topologies coincide for compact manifolds. The proofs which follow, cover the case r is finite. However, all the results hold also for r = ∞, by the way weak and strong C^{∞} topologies are defined [1].

\underline{Proposition 1}. The obvious maps p: C_{π}^{r}(E) → C^{r}(B), p: H_{π}^{r}(E) → H^{r}(B), and p: Diff_{π}^{r}(E) → Diff^{r}(B) are well defined and continuous.

\underline{Proof}. We work with the first map.

To prove that p is well–defined, means that if \overline{f} ∈ C_{π}^{r}(E), then p(\overline{f}) = f ∈ C^{r}(B); but, locally, f = π ∘ \overline{f} ∘ π^{−1}, and so it is C^{r} as a composition of C^{r} maps. Trivially, this implies that f is globally C^{r} and, thus, well defined.

For \overline{f} ∈ C_{π}^{r}(E), let N^{r}(\{U_{i}, ϕ_{i}\}, \{K_{i}\}, \{V_{i}, ψ_{i}\}, \{ϵ_{i}\}, i ∈ Λ) be a basic neighborhood of f [1].

Let us fix a U_{i} from the collection {U_{i}}_{i∈Λ} (this collection is locally finite); let k ∈ K_{i}; then f(k) ∈ V_{i}. We take now e, e_{1} ∈ E such that \overline{f}(e) = e_{1} ∈ π^{−1}(f(k)) and e ∈ π^{−1}(k). There exists a chart Ṽ_{i}^{k} of e_{1} such that π(Ṽ_{i}^{k}) ⊂ V_{i}, and π: Ṽ_{i}^{k} → π(Ṽ_{i}^{k}) is a C^{r} diffeomorphism. We can find a chart W̃_{i}^{k} around e such that \overline{f}(W̃_{i}^{k}) ⊂ V_{i}^{k}, and a chart U_{i}^{k} around k, with a compact neighborhood C_{i}^{k} of k such that C_{i}^{k} ⊂ U_{i}^{k} ⊂ U_{i}, and π(W̃_{i}^{k}) = U_{i}^{k}, with π: W̃_{i}^{k} → U_{i}^{k} a C^{r} diffeomorphism. Then f(U_{i}^{k}) ⊂ π(Ṽ_{i}^{k}). We consider now π^{−1}(C_{i}^{k}) ∩ W̃_{i}^{k}. If we repeat this for each k ∈ K_{i}, we can get a finite cover of K_{i} with U_{i}^{k_{l}} and C_{i}^{k_{l}}, l = 1, 2, …, j_{i}, (C_{i}^{k_{l}} ⊂ U_{i}^{k_{l}}). The collection {W̃_{i}^{k_{l}}}_{i∈Λ,l=1,2,... ,j_{i}} is locally finite as is easily seen, and \overline{f}(π^{−1}(C_{i}^{k_{l}}) ∩ W̃_{i}^{k_{l}}) ⊂ Ṽ_{i}^{k_{l}}. We consider the collection \{ϵ_{i}^{k_{l}}\}_{i∈Λ,l=1,2,... ,j_{i}} of positive numbers, where ϵ_{i}^{k_{l}} = ϵ_{i}. Then, for the basic neighborhood N^{r}({W̃_{i}^{k_{l}}, ϕ_{i} ∘ π\}, \{π^{−1}(C_{i}^{k_{l}}) ∩ W̃_{i}^{k_{l}}\}, \{Ṽ_{i}^{k_{l}}, ψ_{i} ∘ π\}, \{ϵ_{i}^{k_{l}}\}, i ∈ Λ, l = 1, 2, …, j_{i}) of \overline{f}, we have that if \overline{g} belongs to this neighborhood, then g = p(\overline{g}) satisfies: g(K_{i}) ⊂ V_{i}, i ∈ Λ. Let x ∈ (ϕ_{i} ∘π)(π^{−1}(C_{i}^{k_{l}})∩W̃_{i}^{k_{l}}) = ϕ_{i}(C_{i}^{k_{l}}); then, for s = 0, …, r, ∥D^{s}(ψ_{i} ∘π ∘\overline{f} ∘π^{−1} ∘ ϕ_{i}^{−1})(x)−D^{s}(ψ_{i}∘π∘\overline{g}∘π^{−1}∘ϕ_{i}^{−1})(x)∥ < ϵ_{i}. Since, locally, \overline{f} = π^{−1}∘f ∘π, \overline{g} = π^{−1}∘ g ∘ π, the above inequality leads to ∥D^{s}(ψ_{i} ∘ f ∘ ϕ_{i}^{−1})(x) − D^{s}(ψ_{i} ∘ g ∘ ϕ_{i}^{−1})(x)∥ < ϵ_{i}, for s = 0, …, r, x ∈ ϕ_{i}(K_{i}); thus, g ∈ N^{r}({U_{i}, ϕ_{i}}, {K_{i}}, {V_{i}, ψ_{i}}, {ϵ_{i}}, i ∈ Λ), and p is continuous. The Proposition holds for the other two maps as an obvious consequence.

The following Lemma has an easy proof which is omitted.

\underline{Lemma 2}. The set theoretical inclusion i: C^{r}(B) → C(B) (H^{r}(B) → H(B), Diff^{r}(B) → Top(B), respectively) is a continuous map if C^{r}(B) (H^{r}(B), Diff^{r}(B), respectively) is given the weak or the strong C^{r} topology.

For the following results, the spaces are given only the weak C^{r} topology.

\underline{Proposition 3}. C_{π}^{r}(E) is the fibered product of i: C^{r}(B) → C(B), and p: C_{π}(E) → C(B). Similar statements hold for H_{π}^{r}(E), Diff_{π}^{r}(E).

\underline{Proof}. The fibered product of i and p is the subspace U of C^{r}(B) × C_{π}(E) consisting of all pairs (f, \overline{f}) for which f = p(\overline{f}). We now show that the correspondence τ: C_{π}^{r}(E) → U, given by τ(\overline{f}) = (f, \overline{f}) is a homeomorphism. That it is 1–1, is immediate; to show that it is onto, all we have to show is that if f ∈ C^{r}(B) has a lifting \overline{f} ∈ C_{π}(E), then \overline{f} ∈ C_{π}^{r}(E). Since π is a local C^{r} diffeomorphism, locally \overline{f} = π^{−1} ∘ f ∘ π, thus \overline{f} is C^{r} differentiable, and τ is onto. Now τ = p × i, where i is the inclusion C_{π}^{r}(E) → C_{π}(E) which, by an obvious modification of Lemma 2, is continuous. Since p is also continuous (Proposition 1), τ is continuous.

Finally, we have to show that τ^{−1} is continuous. Let N^{r}({U_{i}, ϕ_{i}}, {K_{i}}, {V_{i}, ψ_{i}}, \{ϵ_{i}\}, i ∈ I) be a basic neighborhood of \overline{f} ∈ C_{π}^{r}(E) (here I is finite). We can always assume that π: U_{i} → π(U_{i}), π: V_{i} → π(V_{i}) are C^{r} diffeomorphisms, for each i ∈ I. To see why this is so, let us say that U_{i}, V_{i}, for some i, do not have this property. Then, for each k ∈ K_{i}, consider a chart (W_{k}, ψ_{i}) around \overline{f}(k), with W_{k} ⊂ V_{i} and π: W_{k} → π(W_{k}) being a C^{r} diffeomorphism. By continuity of \overline{f} we may find a compact neighborhood C_{k} of k and a chart (U′_{k}, ϕ_{i}) around k such that C_{k} ⊂ U′_{k} ⊂ U_{i}, \overline{f}(C_{k}) ⊂ W_{k}, and π: U′_{k} → π(U′_{k}) is a C^{r} diffeomorphism. Now K_{i} is covered by a finite number of such C_{k}’s say C_{k_{1}}, C_{k_{2}}, …, C_{k_{j}}. We replace in our neighborhood, U_{i} with {U′_{k_{l}}}_{l=1}^{j}, K_{i} with {C_{k_{l}}}_{l=1}^{j}, V_{i} with {W_{k_{l}}}_{l=1}^{j}, and the positive numbers corresponding to the new sets can all be taken equal to ϵ_{i}. The new neighborhood is obviously a subneighborhood of the old one and the charts satisfy the desired property.

We consider now the following neighborhood of (f, \overline{f}) in C^{r}(B) × C_{π}(E): N^{r}({π(U_{i}), ϕ_{i} ∘ π^{−1}}, {π(K_{i})}, {π(V_{i}), ψ_{i} ∘ π^{−1}}, {ϵ_{i}}, i ∈ I) × (⋂_{i∈I} S(K_{i}, V_{i})).

If (g, \overline{g}) is a member of U which belongs to the above neighborhood, then on one hand \overline{g}(K_{i}) ⊂ V_{i}, and on the other hand, for s = 0, 1, …, r and x ∈ ϕ_{i}(K_{i}), ∥D^{s}(ψ_{i} ∘\overline{f} ∘ϕ_{i}^{−1})(x)−D^{s}(ψ_{i} ∘\overline{g} ∘ϕ_{i}^{−1})(x)∥ = ∥D^{s}(ψ_{i} ∘π^{−1} ∘f ∘π ∘ϕ_{i}^{−1})(x)−D^{s}(ψ_{i} ∘ π^{−1} ∘ g ∘ π ∘ ϕ_{i}^{−1})(x)∥ < ϵ_{i}, since x ∈ ϕ_{i}(K_{i}) = (ϕ_{i} ∘ π^{−1})(π(K_{i})), and locally (on U_{i} or V_{i}) π is a C^{r} diffeomorphism, so that π^{−1} makes sense. That is, \overline{g} belongs to the chosen neighborhood of \overline{f}, and τ^{−1} is continuous.

One should notice that since π has taken a local C^{r} diffeomorphism, it has the unique path lifting property. This, together with Propositions 1, 2, and 3 in [2], and Proposition 3 above gives:

\underline{Proposition 4}. If π: E → B is a fibration (regular), then the maps p: C_{π}^{r}(E) → C^{r}(B), H_{π}^{r}(E) → H^{r}(B), Diff_{π}^{r}(E) → Diff^{r}(B) are all fibrations (regular).

The next theorem follows the C^{r} counterpart of Theorem 5 in [2].

\underline{Theorem 5}. If π: E → B is a regular covering map, then so are the obvious maps p: C_{π}^{r}(E) → p(C_{π}^{r}(E)), H_{π}^{r}(E) → p(H_{π}^{r}(E)), Diff_{π}^{r}(E) → p(Diff_{π}^{r}(E)).

\underline{Proof}. By Theorem 5 in [2], the corresponding map p: C_{π}(E) → p(C_{π}(E)) is a regular covering map. From Proposition 3 above, one immediately sees that C_{π}^{r}(E) is the fibered product of C_{π}(E) → p(C_{π}(E)) and the inclusion p(C_{π}^{r}(E)) → p(C_{π}(E)); this proves the theorem for the C_{π}^{r} case. The other two cases are handled similarly.

Now let π: E → B be a regular fibration. If G(C_{π}^{r}(E)/C^{r}(B)), G(H_{π}^{r}(E)/ H^{r}(B)), G(Diff_{π}^{r}(E)/Diff^{r}(B)) denote the groups of deck transformations of the induced fibrations p of Proposition 4, then we have:

\underline{Proposition 6}. The following diagram is a commutative diagram of groups and group homomorphisms:

G(E/B)
↓^{i}
G(C_{π}(E)/C(B)) ⟶^{r} G(H_{π}(E)/H(B)) ⟶^{r} G(Top_{π}(E)/Top(B))
↓^{r}  ↓^{r}  ↓^{r}
G(C_{π}^{r}(E)/C^{r}(B)) ⟶^{r} G(H_{π}^{r}(E)/H^{r}(B)) ⟶^{r} G(Diff_{π}^{r}(E)/Diff^{r}(B))
↑^{i}
G(E/B)

where r is defined in all cases by restriction to the corresponding subset, and the i’s are group embeddings given by i(\overline{f})(\overline{h}) = \overline{f} ∘ \overline{h}.

\underline{Proof}. The upper half of the diagram is just Proposition 4 in [2]. The horizontal r’s are well–defined homomorphisms by an obvious C^{r} modification of the argument given in the above mentioned proposition. We now show that the vertical r’s are well–defined homomorphisms. If \overline{f} ∈ C_{π}^{r}(E), and σ ∈ G(C_{π}(E)/C(B)), then σ(\overline{f}) ∈ C_{π}^{r}(E) (since p(\overline{f}) ∈ C^{r}(B), all its liftings are members of C_{π}^{r}(E) as we saw in the proof of Proposition 3; but σ(\overline{f}) is such a lifting of f = p(\overline{f}), proving the assertion). That r(σ) is 1–1 and onto is straightforward (repeat similar argument in Proposition 4 in [2]). Continuity of r(σ) follows from the fact that this map is the restriction of the map 1 × σ: C^{r}(B) × C_{π}(E) → C^{r}(B) × C_{π}(E) to the fibered product C_{π}^{r}(E). Finally, (r(σ))^{−1} = r(σ^{−1}), which is continuous; thus, r is well defined. That it is a homomorphism is straightforward. The same proof works for the other two vertical r’s. We now show that the obvious map i: G(E/B) → G(Diff_{π}^{r}(E)/Diff^{r}(B)) is a well–defined group embedding. We first notice that the elements of G(E/B) are all the liftings of the identity 1: B → B; consequently, they are all C^{r} diffeomorphisms. This means that the map \overline{f} ∘ \overline{h} is C^{r} differentiable for \overline{f} ∈ G(E/B), \overline{h} ∈ Diff_{π}^{r}(E). By a similar argument given in Proposition 4 in [2], i(\overline{f}) is 1–1, onto. Continuity of i(\overline{f}) follows from the fact that it is the restriction of 1×i(\overline{f}): Diff^{r}(B)×Top_{π}(E) → Diff^{r}(B)×Top_{π}(E) to the fibered product Diff_{π}^{r}(E). Since i(\overline{f})^{−1} = i(\overline{f}^{−1}), i(\overline{f}) is bicontinuous, thus i is well defined (the same proof works for i: G(E/B) → G(C_{π}^{r}(E)/C^{r}(B)) and i: G(E/B) → G(H_{π}^{r}(E)/H^{r}(B)). The same argument employed in Proposition 4 in [2], proves that i is a group embedding. Commutativity of the whole diagram is straightforward.

\underline{References}

1. M. W. Hirsch, Differential Topology, Springer–Verlag, 1976.

2. M. Magiropoulos, “Induced Fibrations on Spaces of Fiber Transferring Maps,” Missouri Journal of Mathematical Sciences, 12 (2000), 36–41.

Manolis Magiropoulos General Department of School of Technological Applications Technological Educational Institute of Heraklion Stavromenos, Heraklion 71500 Crete, Greece email: mageir@stef.feiher.gr