An note on a small part of the homology of Emd(M, N)

Thomas Schick^{∗} FB Mathematik — Universität Göttingen Bunsenstr. 3 — 37073 Göttingen — Germany

Last compiled February 12, 2002; last edited Feb 12, 2002 or later

Abstract

Let M and N be smooth manifolds and Emd(M, N) the space of embeddings from M to N. If there exist small embeddings of M into N (with image contained in suitable coordinate neighborhoods), we show how elements in the homology of Emd(M, N) can be detected in the homology of N. We consider examples which show that this way one can not get much information about “large” embeddings.

1 The space of small embeddings and its homology

1.1 Definition. Let N be a compact Riemannian manifold of dimension n, r > smaller than the injectivity radius of N (i.e. the exponential map exp_{x}: T_{x}N → N is a diffeomorphism restricted to the ball of radius r).

Let Emb(M, N) be the space of all smooth embeddings from M to N. We equip it with the weak C^{∞}-topology (C^{∞}-convergence on compact subsets of M) (compare [2]). All statements are also valid for the weak C^{r}-topology, if r ≥ 1.

Fix p ∈ M. Let Emb_{r}(M, p; N) the subspace consisting of embeddings i: M → N with image contained in the r-neighborhood of j(p).

Observe that Emb_{r}(M, p; N) is nonempty if and only if M admits an embedding into ℝ^{n}. The case (M, p) = (ℝ^{n}, 0) will be of particular importance.

1.2 Proposition. The following observations are well known or immediately verified.

(1) By definition, we have an embedding

Emb_{r}(M, N) → Emb(M, N).

∗e-mail: schick@uni-math.gwdg.de www: http://www.uni-math.gwdg.de/schick Fax: ++49 -551/39 2985

(2) Fix an embedding j: M_{1} → M_{2}. By pre-composition, this induces continuous maps

Emb(M_{2}, N) → Emb(M_{1}, N) Emb_{r}(M_{2}, N) → Emb_{r}(M_{1}, N).

(3) Fix q ∈ M. Evaluation defines continuous maps

ev_{q}: Emb(M, N) → N; i ↦ i(q),

and similarly for Emb_{r}(M, p; N). In particular, the adjoint of the identity map gives a map

ev: Emb(M, N) × M → N; (i, q) ↦ i(q).

All of this maps are compatible with pre-composition. We get for an embedding j: (M_{1}, p) → (M_{2}, j(p)) e.g. a commutative diagram

Emb_{r}(M_{2}, j(p); N) ⟶ Emb_{r}(M_{1}, p; N)
↓^{ev_{j(q)}}  ↓^{ev_{q}}
N  N.

(4) Fix p ∈ M. Assume dim(M) = m. Then we can consider the bundle V_{m}(T N) := Iso(T_{p}M, T N) over N, with fiber over x ∈ N the linear monomorphisms from T_{p}M to T_{x}N. Choosing a basis of T_{p}M induces between isomorphism of V_{n}(T N) and the bundle of m-frames in T N. Clearly, this is a principal Gl_{m}(ℝ)-bundle.

The above evaluation maps can be lifted to maps

\overline{ev_{p}}: Emb(M, N) → V_{m}(T N); i ↦ (T_{q}i),

and similarly for Emb_{r}(M, N).

(5) Let Map(M, N) be the space of all continuous maps from M to N. Clearly, ev_{p} and ev factor through Map(M, N):

Emb(M, N) ↪ Map(M, N) ⟶^{ev_{p}} N.

The aim of this note is to get some information about the (co)homology of Emb(M, N) in terms of the (co)homology of N and of its frame bundle V_{m}(T N). This way, we generalize the results of [4] and give a more algebraic topological proof of his result.

We will simply consider only the maps

\begin{aligned}(ev_{p})_{∗} & : H_{∗}(Emb(M, N)) → H_{∗}(N), \\ (ev_{p})^{∗} & : H^{∗}(N) → H^{∗}(Emb(M, N)) \end{aligned}

induced by evaluation at p ∈ M. In [4], more generally maps H^{∗}(N) → H^{∗}(Emb(M, N)) was constructed for every continuous linear functional ϕ on C^{∞}(M) (using the de Rham point of view). This was done in the following way:

An note on the homology of Emd(M, N) 3

To a differential form ω ∈ Ω^{r}(N) we assign the differential form ω_{ϕ} on Emb(M, N). At tangent vectors v_{1}, ⋯, v_{r} in T_{i}Emb(M, N) (i.e. vector field of N along the embedding i: M → N) ω_{ϕ}(v_{1}, …, v_{r}) has the value

ϕ(ω(v_{1}, …, v_{r}) ∘ i).

Note that ω(v_{1}, …, v_{r}) is a smooth function on i(M).

Obviously, if ϕ is the δ-function at p, this map is the same as the one induced by ev_{p}. We now show that all these maps are essentially the same.

1.3 Lemma. Let ϕ_{1}, ϕ_{2} ∈ C^{∞}(M)′ be two linear functional which represent the same element in de Rham homology (defined using currents). Then the two different maps H^{∗}(N) → H^{∗}(Emb(M, N)) defined using ϕ_{1} or ϕ_{2} coincide.

Proof. Composition of ev with the slant product gives a map

H^{k}(N) × H_{0}(M) ⟶^{ev×id} H^{K}(Emb(M, N) × M) × H_{0}(M) ⟶^{H}^{k} (Emb(M, N)).

Using de Rham cohomology (with differential forms) and de Rham homology (with currents), the slant product with a zeroth homology class, represented by a functional ϕ on C^{∞}(M), is given exactly in the way described above. This already concludes the proof.

1.4 Corollary. If M is connected, then in the situation of Lemma 1.3 ϕ_{1}(1) = ϕ_{2}(1) implies that the induced maps H^{∗}(N) → H^{∗}(Emb(M, N)) coincide, where 1 is the function with constant value 1.

Proof. If M is connected then H_{0}(M; ℝ) ≅ ℝ ≅ H^{0}(M; ℝ). Two cycles represent the same homology class, if they are equal on the generator 1 of the cohomology group.

From now on, we will always use the functional given by the delta function at p. This way, we can avoid using the slant product, and simply look at the map induced by ev_{p}. Because of the Lemma 1.3 and its proof, this is no loss of generality, since the general case is a linear combination of such functionals.

The main result of [4] was, that under suitable conditions the map H^{1}(N) → H^{1}(Emb(M, N)) is a monomorphism. We will reprove this result and show that its reason is, that the map \overline{ev} has a section.

1.5 Theorem. Set M := ℝ^{k}. Then there is a section

f: V_{k}(T N) → Emb_{r}(M, 0; N)

of \overline{ev_{0}}. In particular, the map induced by ev_{0} in homology is split surjective, and in cohomology it is split injective (with arbitrary coefficients).

Proof. Recall that V_{k}(T N) in this particular case is equal to k-frame bundle of T N, since a linear monomorphism of T_{0}ℝ^{k} = ℝ^{k} to T_{x}N is the same as a frame in T_{x}N. To such a frame we associate the embedding

(ℝ^{k}, 0) ≅ (B_{r}(0) ⊂ ℝ^{n}, 0) → N

given by the exponential map. Our choice of r > 0 implies that this is indeed an embedding. The exponential map depends smoothly on the basepoint, and is defined in such a way that this map is a splitting of ev_{0}.

1.6 Remark. The statement of Theorem 1.5 remains true if N is not compact. In the proof, one simply has to replace the constant r by a smooth function r(x) > 0, such that r(x) is smaller than the injectivity radius of N at x. Of course, the definition of Emb_{r}(M, N) has to be modified accordingly.

Next, we want to get some information about the homology of V_{k}(T N), and in particular compare it with the homology of N. This is done in a standard way, using the fiber bundle V_{k}(ℝ^{n}) → V_{k}(T N) → N, compare [6, 3]. Since V_{k}(ℝ^{n}) is n − k − 1-connected, the bundle projection V_{k}(T N) → N is n − kconnected. In particular, it induces an isomorphism in H_{i} and H^{i} for i < n − k, and an epimorphism in H_{n−k} as well as a monomorphism in H^{n−k}.

An additional refinement is possible if k = n and N is an oriented manifold. In this case we can look at the subspace V_{n}^{o}(T N) of V_{n}(T N) of oriented frames (corresponding to the subspace Emb_{r}^{o}(ℝ^{n}, N) of orientation preserving embeddings). The fiber of V_{n}^{o}(T N) is isomorphic to Gl_{n}^{+}(ℝ), the group of orientation preserving linear isomorphisms of ℝ^{n}. This group is homotopy equivalent to SO(n, ℝ). In particular, it is connected. Consequently, the above argument implies that the map V_{n}^{o}(T N) → N induces an epimorphism in H_{1} and a monomorphism in H^{1}.

Combining this information with the information we get from the split of \overline{ev_{0}} : Emb_{r}(ℝ^{k}, 0; N) → V_{k}(T N), we see that ev_{0} : Emb_{r}(ℝ^{k}, 0; N) → N induces a epimorphism on H_{i}, and a monomorphism on H^{i}, for i ≤ n − k.

In the special case k = n and N orientable, the statement is true for i ≤ 1. The latter result follows from the commutative diagram

Emb_{r}^{o}(ℝ^{n}, 0; N) ⟶^{\overline{ev_{0}}} V_{n}^{o}(T N) ⟶ N
↓  ↓  ↓^{=}
Emb_{r}(ℝ^{n}, 0; N) ⟶^{\overline{ev_{0}}} V_{n}(T N) ⟶ N.

Assume V_{k}(T N) → N has a split, i.e. T N has a trivial k-dimensional subbundle. Then, the above considerations imply that ev_{0} : Emb_{r}(ℝ^{k}, 0; N) → N induces a split epimorphism on H^{i}, and a monomorphism on H^{i}, for all i ∈ ℕ. In particular, this is true for k = n if N is parallelizable.

Assume M is an m-dimensional manifold which admits an embedding j: M ↪ ℝ^{k}. Fix a basepoint p ∈ M and assume j(p) = 0 (this is no loss of generality). This induces a particular monomorphism T_{p}j: T_{p}M → T_{0}ℝ^{k} = ℝ^{k}, which induces by pre-composition a map V_{k}(T N) → V_{m}(T N) (here it is most convenient, to define V_{k} using embeddings of ℝ^{k}, and V_{m} using embeddings of T_{p}M). This gives a commutative diagram

Emb_{r}(ℝ^{k}, 0; N) ⟶^{\overline{ev_{0}}} V_{k}(T N) ⟶ N
↓  ↓  ↓^{=}
Emb_{r}(M, p; N) ⟶^{\overline{ev_{p}}} V_{m}(T N) ⟶ N.

Note that the splitting of \overline{ev_{0}} does not provide us with a splitting of \overline{ev_{p}}. However, the map immediately implies that ev_{p}: Emb_{r}(M, p; N) → N induces an epimorphism on H_{i} in the same range where ev_{0} : Emb_{r}(ℝ^{k}, p; N) does (and similarly for the dual statement in cohomology).

Last, observe that ev_{p} factors through Emb(M, N). Therefore, we have proved the following theorem.

1.7 Theorem. Assume M admits an embedding into ℝ^{k}. Then we have an epimorphism

H_{i}(Emb(M, N)) → H_{i}(N) for i ≤ n − k.

If k = n and N is orientable, we get an epimorphism

H_{1}(Emb(M, N)) → H_{1}(N).

The dual maps in cohomology are monomorphisms in the same range.

If T N has a k-dimensional trivial sub-bundle, the above result holds for all i ∈ ℕ. In particular, this is the case if k = n and N is parallelizable.

In the same way, we get further information (in the range i ≤ l) if the restriction of T N splits of a k-dimensional trivial sub-bundle. This condition can be replaced by assuming that the restriction of T N to any l-dimensional immersed submanifold splits of a k-dimensional trivial bundle, as long as we work with rational or real coefficients.

1.1 Reverse implications for T N

We have just seen that N being parallelizable implies that ev_{p}: Emb_{r}(M, p; N) → N induces a split surjection in homology, and a split injection in cohomology.

On the other hand, note that the pull back of T N to V_{m}(T N) canonically splits off a trivial m-dimensional sub-bundle V_{M}(T N) × T_{p}M, and the same follows then for (ev_{p})^{∗}T N on Emb(M, p; N). Consequently, if ev_{p} induces an injection in homology (in a given range of degrees) the characteristic cohomology classes of T N look (in the given range) like the characteristic classes of a bundle splitting off a trivial m-dimensional sub-bundle. In particular, the Pontryagin classes p_{i}(T N) vanish for 4i > n−m (in the given range), and, if N is orientable, the Euler class e(T N) vanishes if m > 0 and n is in the given range.

1.8 Remark. Note that we information about H_{∗}(Emb(M, N)) all comes from the subspace of small embeddings Emb_{r}(M, N). In particular, there might be many components of Emb(M, N) we don’t get any information about. We will discuss this a little bit in the next section.

1.9 Remark. Let Emb^{0}(M, N) be the space of embeddings with the C^{0}-topology. The identity map is continuous from Emb(M, N) with the C^{∞}-topology to Emb^{0}(M, N) and gives a factorization of the evaluation map. Therefore, the statements of Theorem 1.5 remain true with Emb(M, N) replaced by Emb^{0}(M, N).

It is conceivable that some of the statements remain true with the space of smooth embeddings replaced by the space of continuous embeddings. However, the proof given here relies on the smooth structure.

1.10 Remark. We can deal with homology as well as cohomology, and that not only real, but integer coefficients are allowed, in contrast to the treatment of

1.2 Fibrations

1.11 Theorem. The evaluation map

ev_{p}: Emb_{r}(M, p; N) → N

is a locally trivial fiber bundle with typical fiber the space Emb(M, ℝ^{n}) of embeddings of i: M → ℝ^{n} with i(p) = 0.

Proof. Fix x_{0} ∈ N and ϵ > 0 small enough. Choose a Riemannian trivialization τ of T N|_{B_{ϵ}(x_{0})} (it could e.g. be given by geodesic coordinates at x_{0}).

Now we define a smooth family of diffeomorphisms ϕ_{x}: B_{r}(x) → B_{r}(x_{0}), where ϕ_{x} is the composition of exp_{x}^{−1} with the map between T_{x}N and T_{x_{0}}N given by the trivialization and then with exp_{x_{0}}.

Set Emb_{r}(M, p; N, x_{0}) := {i ∈ Emb_{r}(M, p; N) | i(p) = x_{0}}. We now get a trivialization

\begin{matrix}Emb_{r}(M, p; N)|_{B_{r}(x_{0})} & → B_{r}(x_{0}) × Emb_{r}(M, p; N, x_{0}) \\ i & ↦ (i(p), ϕ_{i(p)} ∘ i). \end{matrix}

1.12 Question and Remark. Is it true that also \overline{ev_{p}}: Emb_{r}(M, p; N) → V_{m}(N) is a locally trivial fiber bundle?

The method of Theorem 1.11 applies to prove this result if we restrict to the subspaces of “metric” embeddings i: M → B_{r}(i(p)) such that T i: T_{p}M → T_{i(p)}N is an isometric embedding (for a fixed, but arbitrary Riemannian metric on T_{p}M).

A slight modification implies that Emb_{r}(M, p; N) → V_{m}(N) is a local fibration (i.e. has locally the homotopy lifting property): we have to lift a homotopy X × [0, 1] → U ⊂ V_{m}(N) to Emb_{r}(M, p; N). When we try to trivialize Emb_{r}(M, p; N) → V_{m}(N) over a neighborhood U of an m-frame α_{0} as in the proof of Theorem 1.11, we have to use an additional (intermediate) linear isomorphism B_{α}: T_{x_{0}}N → T_{x_{0}}N which maps the (image of the) m-frame α ∈ U we start with to the m-frame α_{0}. If (the image of) α is close to α_{0} we define this linear map by mapping the l-th vector of α to the l-th vector of α_{0}, and doing nothing on the complement of the span of α (if α is close enough to α_{0}, this is an isomorphism).

The problem is that this map need not be an isometry, and consequently that we don’t create a diffeomorphism B_{r}(x) → B_{r}(x_{0}).

Let

v(α, β) := sup{|B_{α}^{−1}B_{β}v| | 0 ≠ v ∈ T_{x_{0}}N}

be the maximal distortion for the corresponding “linear part” of the map from the fiber over β ∈ U to the fiber over α ∈ U.

To lift the homotopy h: X × [0, 1] → U, we use the “trivializations”, but compose them with (nonlinear) “contractions” of T_{x_{0}}N which are the identity near zero, but counteract the distortion v(h(x), x) which would prevent our lift from living in Emb_{r}(M, p; N).

Since N and V_{m}(N) are paracompact and Hausdorff, any local fibration, in particular any locally trivial fiber bundle, are fibrations, and we can apply the long exact homotopy sequence of the fibration. Consequently, we get long exact sequences

⋯ → π_{k}(Emb_{r}(M, p; ℝ^{n}, 0)) → π_{k}(Emb_{r}(M, p; N)) → π_{k}(N) → ⋯

⋯ → π_{k}(Emb_{r}^{f}(M, p; ℝ^{n}, 0)) → π_{k}(Emb_{r}(M, p; N)) → π_{k}(V_{m}(T N)) → ⋯

where Emb_{r}^{f}(M, p; ℝ^{n}, 0) is the space of embeddings with prescribed differential at p which we encountered earlier. Observe that the \overline{ev_{p}}: Emb_{r}(M, p; N) → V_{m}(N) has a split, such that the second long exact sequence splits into short exact sequences

0 → π_{k}(Emb_{r}^{f}(M, p; ℝ^{n}, 0)) → π_{k}(Emb_{r}(M, p; N)) → π_{k}(V_{m}(T N)) → 0.

Note, however, that the space of embeddings into ℝ^{n} is a very complicated space. But we can see at last that we can split off the homotopy groups of V_{m}(T N), which are closely related to the homotopy groups of N.

2 Non-small embeddings

Here, we want to show that the evaluation map does not give much information about the homology of components of Emb(M, N) where the embeddings are not contained in small balls.

We use the simple observation that ev_{p}: Emb(M, N) → N factors through the space Map(M, N) of all continuous maps from M to N. Consequently, if (ev_{p})_{∗}: H_{∗}(Map_{i}(M, N)) → H_{∗}(M) is not surjective, neither can the map from H_{∗}(Emb_{i}(M, N)) be surjective, where Emb_{i}(M, N) is the component of i: M ↪ N in Emb(M, N), and similarly for Map_{i}(M, N).

Note that ev_{p}: Map(M, N) → N is always a fibration (as long as the inclusion p ↪ M is a cofibration, which is the case for manifolds), compare [5, 2.8.2].

Now, we specialize to M = S^{1} and N a connected orientable surface of genus > 1. We claim that Map(S^{1}, N) has many components which contain embeddings i: S^{1} → N, such that (ev_{p})_{∗}: H_{1}(Map_{i}(S^{1}, N)) → H_{1}(N) is not surjective.

To see this, we examine the long exact sequence in homotopy of the fibration Ω_{x_{0}}N → Map(S^{1}, N) → N, where Ω_{x_{0}}N is the space of loops in N based at x_{0}. Recall that N is aspherical, which implies that π_{k}(Ω_{x_{0}}N) = 0 for k > 0, and π_{0}(Ω_{x_{0}}N) = π_{1}(N, x_{0}) (the isomorphism is canonical, mapping [γ] ∈ π_{1}(N, x_{0}) to [γ] ∈ π_{0}(Ω_{x_{0}}N)).

As the relevant part of the long exact sequence of this fibration we obtain therefore for any basepoint i ∈ Ω_{x_{0}}N

0 → π_{1}(Map(S^{1}, N), i) ⟶^{(ev_{p})_{∗}} π_{1}(N, x_{0}) ⟶^{δ} π_{1}(N, x_{0}) → π_{0}(Map(S^{1}, N)) → {∗}

(π_{0}(N) consists of a single point since N is connected).

Usually, this sequence is considered when i is the constant loop. However, it is valid for any loop i, and we want to investigate it when i represents some interesting elements in π_{1}(N, x_{0}).

Obviously, to understand the map (ev_{p})_{∗}, we have to understand the boundary map δ. In this situation, δ maps α ∈ π_{1}(N, x_{0}) to α^{−1}[i]α ∈ π_{1}(N, x_{0}) (this follows easily from the definition, where δ is induced from the effect of fiber transport to the basepoint i).

This sequence being exact at the points involving π_{0} (which is only a set, not a group) simply means that the image of (ev_{p})_{∗} is exactly the subgroup which is mapped to the basepoint [i] under the map δ. Our calculation shows that this is exactly the centralizer of [i].

It is known [1, Section 1 and Theorem 3] that the centralizer of each nontrivial element of π_{1}(N) is cyclic. If [i] ∈ π_{1}(N) is primitive (i.e. not a proper power) it therefore is the cyclic subgroup generated by [i].

In particular, the map induced by ev_{p} is far from being surjective, here. Passing to the abelianization, we get the map

(ev_{p})_{∗}: H_{1}(Map_{i}(S^{1}, N)) → H_{1}(N).

Note that we obtain H_{1} of the component of Map(S^{1}, N) containing i. Our considerations show that the image is contained in the image of a cyclic group in the abelianization. Therefore, (ev_{p})_{∗} is not surjective.

2.1 Question. When is ev_{p}: Emb(M, N) → N a fibration?

References

[1] Gilbert Baumslag. On generalised free products. Math. Z., 78:423–438, 1962.

[2] Morris W. Hirsch. Differential topology. Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original.

[3] John W. Milnor and James D. Stasheff. Characteristic classes. Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76.

[4] Olaf Müller. A note on H^{1}(Emb(M, N)). arXiv: http://de.arXiv.org/abs/math.AT/0202049, 2002.

[5] Edwin H. Spanier. Algebraic topology. Springer-Verlag, New York, 1981. Corrected reprint.

[6] Norman Steenrod. The topology of fibre bundles. Princeton University Press, Princeton, NJ, 1999. Reprint of the 1957 edition, Princeton Paperbacks.