HOMOTOPY CLASSES WITH SMALL JACOBIANS

STEFAN BECHTLUFT-SACHS

Abstract

If the infimum of the conformal k-Jacobian on the homotopy class of a map between compact Riemannian manifolds vanishes then the map factors rationally through the k-skeleton of the target manifold.

AMS Subject Classification: 58E15 (53C43 57D99)

1. Introduction

It follows from the Sobolev inequality that a map f: M^{m} → X between connected compact Riemannian manifolds M and X, m = dim(M), is nullhomotopic if its differential df has sufficiently small L^{r}-norm for some r > m. In fact, the diameter of the image of f is bounded by diam(im(f)) ≤ C∥df∥_{r} with some constant C depending only on M, X and r > m. In the conformal case r = m this simple argument fails. A map f with arbitrary small ∥df∥_{m} can have arbitrarily large image. But by a theorem of White [10] there is a constant ϵ > 0 depending on the geometries of M and X such that f is nullhomotopic if ∥df∥_{m} < ϵ.

We consider the analogous question for the Jacobian in place of the L^{r}-norm. For k ∈ ℕ and r ∈ ℝ^{+} these are the functionals

(1)  J_{k}^{r}: C^{∞}(M, X) → ℝ_{0}^{+} , J_{k}^{r}(f) = ∫_{M} ϕ(df)

where

ϕ(df) = | \underbrace{df ∧ … ∧ df}_{k} |^{r/k} = σ(df^{∗}df)^{r/2k}

and σ_{k}(df^{∗}df) denotes the kth elementary symmetric polynomial in the eigenvalues of df^{∗}df. If r = m = dim(M) this functional is invariant under conformal changes of the metric on M.

In more general framework, for functionals E: C^{∞}(M^{m}, X) → ℝ_{0}^{+} we are interested in the information on the homotopy class of f detected by the infimum

Ẽ: [M^{m}, X] → ℝ_{0}^{+} , Ẽ(f) := inf{E(g)|g: M^{m} → X, g ≃ f} ,

in particular in the consequences of Ẽ(f) = 0.

We write E_{1} >> E_{2} if lim_{ν} E_{1}(f_{ν}) = 0 implies lim_{ν} E_{2}(f_{ν}) = 0 for any sequence (f_{ν})_{ν} in C^{∞}(M^{m}, X). Among the Jacobians the Hölder inequality gives estimates

(2)  J_{1}^{r} >> J_{2}^{r} >> ⋯ J_{l}^{r} >> ⋯ >> J_{m}^{r}

and J_{l}^{r_{1}} >> J_{l}^{r_{2}} if r_{1} ≥ r_{2}. With respect to >>, the Jacobian J_{1}^{r}(f) is equivalent to the L^{r}-norm of the differential and J_{m}^{m} >> vol(im f). If f is homotopic to a map f̃: M → X^{l−1} ⊂ X into the (l − 1)-skeleton X^{l−1} of a triangulation of X we obviously have J̃_{l}^{r}(f) = 0 for any r. The converse is known to hold in the extreme cases l = 1 and l = m of (2) if r ≥ m. It follows from a theorem of Pluzhnikov, [7], and White, [10], that J̃_{l}^{r} depends only on the restriction of f to the [r]-skeleton of a triangulation of M.

Let f: M^{m} → X be a map with J̃_{l}^{m}(f) = 0. In [2] it is shown that f behaves homologically like a map into the (l − 1)-skeleton X^{l−1} of a triangulation of X, i.e. induces 0 in homology of degree at least l. It is also shown there that f does not need to be homotopic to a map into X^{l−1}. Counterexamples produced in [2] arise from torsion elements in the higher homotopy groups of spheres. This suggests that f factors rationally. We prove:

Theorem 3. Let X^{l−1} be the (l − 1)-skeleton of a triangulation of X and assume that π_{1}(X^{l−1}) = 0. Let f: M^{m} → X be a map with J̃_{l}^{m}(f) = 0. Then the rationalization f_{ℚ}: M → X_{ℚ} is homotopic to a map f_{ℚ}: M → X_{ℚ}^{l−1} into the rationalization of the (l − 1)-skeleton of X.

Remarks

(1) Theorem 3 extends a result of Rivière in [8] who showed that the Hopf invariant of a map f: S^{4k−1} → S^{2k} is estimated by J̃_{2k}^{4k−1}. For maps between spheres the rational homotopy type is controlled by the Hopf invariant.

(2) For the Jacobians J_{l}^{r}(f) with l ≥ 2 and arbitrary large r one easily constructs surjective maps M^{m} → X^{l} with J̃_{l}^{m}(f) = 0. Thus a simple argument based on a Sobolev-type inequality is not available in this case.

2. Factorization in Rational Homotopy

The proof of Theorem 3 is a computation in suitable relative Sullivan algebras, along the lines of [5], [6] where the number of homotopy classes of maps f was estimated by bounds on the dilatation. As before X^{l−1} denotes the (l − 1)- skeleton of X. We denote by X_{ℚ}, X_{ℚ}^{l−1} the rationalisations of X and X^{l−1} respectively. Thus we have maps X → X_{ℚ} and X^{l−1} → X_{ℚ}^{l−1} inducing isomorphisms H^{∗}(X, ℚ) ≅ H^{∗}(X_{ℚ}, ℤ) and H^{∗}(X^{l−1}, ℚ) ≅ H^{∗}(X_{ℚ}^{l−1}, ℤ). We assume that X^{l−1} is simply connected and l ≥ 2. Then X is also simply connected and the above rationalisations are unique up to homotopy.

Let Ω(M), Ω(X) and Ω(X^{k}) denote the respective algebras of differential forms. By the functorial properties of Sullivan algebras the rationalization f_{ℚ}: M → X_{ℚ} is homotopic to a map F: M → X_{ℚ}^{l−1} ⊂ X_{ℚ} if there is a relative Sullivan algebra S := Ω(X) ⊗_{d} ΛV ≃ Ω(X^{l−1}) and an extension F^{∗}: S → Ω(M) of f^{∗}: Ω(X) → Ω(M), see [3], [4]. We will first construct a suitable Sullivan algebra S and then use the estimate on the Jacobian of f to define F^{∗}.

2.1. Construction of S

We abbreviate X := Ω(X), Y := Ω(X^{l−1}) and let j: X → Y be the morphism of commutative cochain algebras obtained by restriction. Up to homotopy we want to replace j by a morphism j̃ into a relative Sullivan algebra S homotopy equivalent to Y such that the triangle

\xymatrix{X \ar[r]^{j̃} \ar[rr]_{j} & S = X ⊗_{d} ΛV \ar[r]^{m}_{≃} & Y}

commutes up to homotopy. A relative Sullivan algebra ([4]) is a commutative cochain algebra S = X ⊗_{d} ΛV such that there are graded vector spaces V_{i}, i ∈ ℕ,

V := ⨁_{i>0} V_{i} = ⋃_{i>0} V (i) , V (i) := V (i − 1) ⊕ V_{i} , V (−1) := 0

and homomorphisms

d_{i}: V_{i} → S(i − 1) := X ⊗_{d} ΛV (i − 1)

extending to a differential d: S → S which is nilpotent in the sense that

dV (i) ⊂ S(i − 1) := X ⊗_{d} ΛV (i − 1) where S(−1) := X .

The algebras S(q) are constructed together with morphisms m_{q}: S(q) → Y which are quasiisomorphisms in degrees increasing with q. To begin, let V (0) ↪^{ι} ker d ⊂ Y be a vectorspace of cycles in Y such that V (0)⊕im Hj generates H(Y) as an algebra. Define

S(0) := X ⊗ ΛV (0) with d|_{V (0)} = 0

and

\xymatrix{m_{0} : X ⊗ ΛV (0) \ar[r]^{j⊗Λι} & Y .}

By construction m_{0} induces an epimorphism Hm_{0} : H(X) ⊗ ΛV (0) → H(Y) in homology.

Proceeding by induction in q, assume V (q) and

\xymatrix{S(q) := X ⊗_{d} ΛV (q) \ar[r]^{m_{q}} & Y}

have already been constructed such that Hm_{q} is surjective. Let V_{q+1} ⊂ S(q) with V_{q+1} ≅ ker Hm_{q} ⊂ H(S(q)) be a vector space of representing cycles where the degree on V_{q+1} is set to be the degree inherited from of S(q) diminished by 1. The differential d: V_{q+1} → S(q) is defined to be the inclusion V_{q+1} ↪ S(q). The map m_{q+1} : V_{q+1} → Y is a lift of m_{q} ∘ d over d^{Y}:

(1)  \xymatrix{ker Hm_{q} \ar@{^{(}->}[r] & HS(q) \ar[r]^{Hm_{q}} \ar@{^{(}->}[d] & HY \ar@{^{(}->}[d] \\ V_{q+1} \ar@{-->}[drr]_{m_{q+1}} \ar@{^{(}->}[r]^{d_{q+1}} \ar[u]^{≅} & S(q) \ar[r]^{m_{q}} & Y \\  &  & Y \ar[u]_{d^{Y}}}

Extending m_{q+1} to ΛV (q + 1) = Λ(V_{q+1} ⊕ V (q)) we obtain m_{q+1} : S(q + 1) := X ⊗_{d} V (q + 1) → Y which again induces an epimorphism in homology.

2.2. Extension of f

Let f^{∗}: X = Ω(X) → M = Ω(M) and j^{∗}: X = Ω(X) → Y = Ω(X^{l−1}) be the morphisms of commutative cochain algebras induced by f: M → X and the inclusion j: X^{l−1} ↪ X respectively. The homomorphism induced by j in cohomology is an isomorphism in degrees < l − 1, injective in degree l − 1 and 0 in degrees > l − 1. Hence we may choose V (0) ⊂ H^{l−1}(Y) such that

H(Y) = V (0) ⊕ im Hj .

The morphism m_{0} constructed as before then induces an isomorphism Hm_{0} in degrees ≤ l − 1. Denote by 𝔦(q) the ideal generated by V (0) in S(q). In the diagram (1) we may split

V_{q+1} = V′_{q+1} ⊕ d_{q+1}^{−1}𝔦(q)

where V′_{q+1} lies in degrees > l − 1.

We will inductively extend f^{∗} to maps f_{q}^{∗}: S(q) → M such that the f_{q}^{∗} vanish on 𝔦(q). To this end set

f_{−1}^{∗} := f: S(−1) = X → M

and assume that we already have constructed the extension

f_{q}^{∗}: S(q) → M

satisfying estimates

(2)  ∥f_{q}^{∗}ω∥_{n/r} ≤ ϵ∥ω∥_{n/r}

for all ω ∈ S(q) of degree r, n ≥ r > l − 1. Changing f by a homotopy we can have (2) with arbitrarily small value of ϵ.

For any r-cycle σ in M we find a homologous r-cycle σ′ such that ∫_{σ′} |f_{q}^{∗}ω| < C_{1} ∫_{M} |f_{q}^{∗}ω| where C_{1} does not depend on f_{q}^{∗} (see [10], Proposition 3.1 for instance, or [2], proof of Theorem 3.2). If dω = 0 the Hölder inequality gives an estimate

|∫_{σ} f_{q}^{∗}ω| = |∫_{σ′} f_{q}^{∗}ω| ≤ C_{1}∥f_{q}^{∗}ω∥_{1} ≤ C_{1}C_{2}∥f_{q}^{∗}ω∥_{n/r} ≤ C_{1}C_{2}ϵ∥ω∥_{n/r}

with C_{2} independent of f. In particular f_{q}^{∗}ω is exact if dω = 0.

Thus f_{q}^{∗}(dV′_{q+1}) ⊂ dM. Let {ω_{j}}_{j} be a basis for V′_{q+1}. We define f_{q+1}^{∗} on V_{q+1} by lifting f_{q}^{∗}. More precisely, choose for each ω_{j} some α_{j} ∈ M satisfying dα_{j} = f_{q}^{∗}(dω_{j}) and the estimate (4) of the following Lemma 3. Define f_{q+1}^{∗}|_{𝔦(q)} := 0 and f_{q+1}^{∗}(ω_{j}) := α_{j}.

Lemma 3. Let M be a compact n-dimensional Riemannian manifold and denote by ∥·∥_{p} the L^{p}-norm of differential forms given by the Riemannian metric. There is a constant C ∈ ℝ depending on M (but not on β) such that for each exact rform β ∈ Ω^{r}(M), β ∈ dΩ^{r−1}(M) there is α ∈ Ω^{r−1}(M) with

(4)  β = dα and ∥α∥_{n/(r−1)} ≤ C∥β∥_{n/r} .

Proof: From the Hodge decomposition

Ω(M) = ker Δ ⊕ im d ⊕ im d^{∗}

the Laplacian Δ is invertible on im d ⊕ im d^{∗}. Let α := d^{∗}Δ^{−1}β. Clearly dα = β. Extending the operator Δ^{−1} on im d⊕im d^{∗} by 0 to all of Ω^{r}(M) yields a bounded operator L^{p} = W^{0,p} → W^{2,p} into the Sobolev space W^{2,p}, [9]. Also d^{∗}: W^{2,p} → W^{1,p} is bounded, [1]. From the Sobelev-embedding ι: W^{1,p} ⊂ L^{np/(n−p)} we infer that

∥α∥_{np/(n−p)} ≤ C∥β∥_{p}

where C := ∥ι d^{∗}Δ^{−1}∥_{op} is the operator norm. With p = n/r, np/(n − p) = n/(r − 1) the assertion follows. •

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