On Holomorphic Contractibility of Teichmüller Spaces

: The problem of the holomorphic contractibility of Teichmüller spaces T ( 0, n ) of the punctured spheres ( n > 4) arose in the 1970s in connection with solving algebraic equations in Banach algebras. Recently it was solved by the author. In the present paper, we give a reﬁned proof of the holomorphic contractibility for all spaces T ( 0, n ) , n > 4 and provide two independent proofs of holomorphic contractibility for low-dimensional Teichmüller spaces, which has intrinsic interest

1. PREAMBLE 1.1.A complex Banach manifold X is contractible to its point x 0 if there exists a continuous map F : X × [0, 1] → X with F (x, 0) = x and F (x, 1) = x 0 for all x ∈ X.If the map F can be chosen so that for every t ∈ [0, 1] the map F t : x → F (x, t) of X to itself is holomorphic and F t (x 0 ) = x 0 , then X is called holomorphically contractible to x 0 .
The problem of holomorphic contractibilty of Teichmüller spaces T(0, n) of the punctures spheres (n > 4) arose in the 1970s in connection with solving the algebraic equations in Banach algebras.It was caused by the fact that in the space C m , m > 1, there are domains (even bounded), which are only topologically but not holomorphically contractible (see [7], [8], [15], [16]).
Recently this problem was solved positively in [12].There is established that all space T(0, n), n > 4, are holomorphically contractible.
The proof of Lemma 3 in this paper contains a wrongly assertion (which does not be used there) that the map s m (giving the inclusion of the space T (Γ 0 ) into T (Γ m 0 )) is a section of the forgetful map χ m : T (X m a 0 ) → T (X a 0 ).Such sections do not exist if n > 6.In fact, there was only used in the proof that s m is an open holomorphic map (of a domain onto manifold), and the openness is preserved for the limit map s = lim m→∞ s m which determines an (n − 3)-dimensional complex submanifold s(T (X a 0 )) in the universal Teichmüller space T.
In the present paper, we improve the statement of the indicated Lemma 3 (without changing other arguments in [12] concerning the contractibility of T(0, n)).In the second part of the paper, we provide an alternate proof of holomorphic contractibility of low dimensional Teichmüller spaces (of dimensions two and three), which has an independent interest in view of importance of the problem.The underlying idea of this proof is different from [12].
It remains the Teichmüller space T(1, 3) of tori with three punctures also having dimension three; it will not be involved here.
Another canonical model of T(0, n) = T(X a 0 ) is obtained using the uniformization of Riemann surfaces and the holomorphic Bers embedding of Teichmüller spaces.Consider the upper and lower half-planes U = {z = x + iy : y > 0}, U * = {z ∈ C : y < 0} The holomorphic universal covering map h : U → X a 0 provides is a torsion free Fuchsian group Γ 0 of the first kind acting discontinuously on U ∪ U * , and the surface X a 0 is represented (up to conformal equivalence) as the quotient space U/Γ 0 .The functions µ ∈ L ∞ (X a 0 ) = L ∞ (C) are lifted to U as the Beltrami (−1, 1)-measurable forms µdz/dz on U with respect to Γ 0 satisfying ( µ • γ)γ ′ /γ ′ = µ, γ ∈ Γ 0 and form the corresponding Banach space L ∞ (U, Γ 0 ).We extend these µ by zero to U * and consider the unit ball Belt(U, Γ 0 ) 1 of this space L ∞ (U, Γ 0 ).The corresponding quasiconformal maps w µ are conformal on the half-plane U * , and their Schwarzian derivatives fill a bounded domain in the complex (n−3)-dimensional space B(Γ 0 ) = B(U * , Γ 0 ) of hyperbolically bounded holomorphic Γ 0 -automorphic forms of degree −4 on U * (i.e., satisfy (ϕ This domain models the Teichmüller space T(Γ 0 ) of the group Γ 0 .It is canonically isomorphic to the space T(X a 0 ) (and to a bounded domain in the complex Euclidean space C n−3 ).
Note also that T(Γ 0 ) = T ∩ B(Γ 0 ), where T is the universal Teichmüller space (modelled as domain of the Schwarzian derivatives of all univalent functions on U * admitting quasiconformal extension to U ).

2.2.
The collections (1) fill a domain D n in C n−3 obtained by deleting from this space the hyperplanes {z = (z 1 , . . ., z n−3 ) : z j = z l , j = l}, and with z 1 = 0, z 2 = 1.This domain represents the Torelli space of the spheres X a and is covered by T(0, n), which is given by the following lemma (cf.e.g., [10]; [13], Section 2.8).
Lemma 1.The holomorphic universal covering space of D n is the Teichmüller space T(0, n).This means that for each punctured sphere X a there is a holomorphic universal covering The covering map π a is well defined by where φ a denotes the canonical projection of the ball Belt(C) 1 onto the space T(X a ).
Lemma 1 yields also that the truncated collections a * = (a 1 , . . ., a n−3 ) provide the local complex coordinates on the space T(0, n) and define its complex structure.
These coordinates are simply connected with the Bers local complex coordinates on T(0, n) (related to basises of the tangent spaces to T(0, n) at its points, see [1]) via standard variation of quasiconformal maps of with uniform estimate of the ratio O( µ 2 ∞ )/ µ 2 ∞ on compacts in C (see e.g., [11]).
It turns out that one can obtain the whole space T(X a 0 ) using only the similar equivalence classes [µ] of the Beltrami coefficients from the ball µ ∈ Belt(U ) 1 (vanishing on U * ), requiring that the corresponding quasiconformal homeomorphisms w µ are homotopic on the punctured sphere X a 0 .Surjectivity of this holomorphic quotient map It follows that the function f given by Lemma 2 actually is holomorphic and univalent (hence, maps conformally) in a broader disk D r , r > 1.
First of all, f ′ (z) = 0 on the unit circle S 1 , Indeed, were f ′ (z 0 ) = 0 at some point Now, assuming, in the contrary, that f is not globally univalent in any admissible disk D r with r > 1, one obtains the distinct sequences {z ) for any n, whose limit points z ′ 0 , z ′′ 0 lie on S 1 .Then also in the limit, f (z ′ 0 ) = f (z ′′ 0 ), which in the case z ′ 0 = z ′′ 0 contradicts to univalence of f on S 1 given by Lemma 2 and in the case z ′ 0 = z ′′ 0 = z 0 contradicts to local univalence of f in a neighborhood of z 0 .
The interpolating function f given by Lemma 2 is extended quasiconformally to the whole sphere C across any circle {|z| = r} with r > 1 indicated above.Hence, given a cyclically ordered collection (z 1 , . . ., z m ) of points on S 1 , then for any ordered collection (ζ 1 , . . ., ζ m ) in C, there exists a quasiconformal homeomorphism f of the sphere C carrying the points z j to ζ j , j = 1, . . ., m, and such that its restriction to the closed disk D is biholomorphic on D.
Taking the quasicircles L passing through the points ζ 1 , . . ., ζ m and such that each ζ j belongs to an analytic subarc of L, one obtains quasiconformal extensions of the interpolating function f , which are biholomorphic on the union of D and some neighborhoods of the initial points z 1 , . . ., z m ∈ S 1 .Now Lemma 1 provides quasiconformal extensions of f lying in prescribed homotopy classes of homeomorphisms X z → X w .
Due to the Bers isomorphism theorem, the space T (X ′ a 0 ) is biholomorphically isomorphic to the Bers fiber space [2]).This fiber space is a bounded hyperbolic domain in B(Γ 0 ) × C and represents the collection of domains D µ = w µ (U ) (the universal covers of the surfaces X a 0 ) as a holomorphic family over the space T(0, n − 1) = T (X a 0 ).
The indicated isomorphism between T(0, n + 1) and F(0, n) is induced by the inclusion map j : D * ֒→ D forgetting the puncture at a 0 n , via where j is the lift of j to U and a 0 n−3 is one of the points from the fiber over a 0 n under the quotient map U → U/Γ 0 .
Note also that the holomorphic universal covering maps h : U * → U * /Γ 0 and h ′ : U * → U * /Γ ′ 0 (and similarly the corresponding covering maps in U ) are related by where j is the lift of j.This induces a surjective homomorphism of the covering groups θ : which projects to the surfaces X a 0 and X ′ a 0 as the inclusion of the space Q(X a 0 ) of holomorphic quadratic differentials corresponding to B(Γ 0 ) into the space Q(X ′ a 0 ) (cf. [5]).The Bers theorem is valid for Teichmüller space T(X 0 \ {x 0 }) of any punctured hyperbolic Riemann surface X 0 \ {x 0 } and implies that T(X 0 \ {x 0 }) is biholomorphically isomorphic to the Bers fiber space F(T(X 0 )) over T(X 0 ).
In the exceptional cases of T(1, 2) and T(2, 0), there is a group Γ ′ which contains Γ as a subgroup of index two.Then T(Γ ′ ) = T(Γ), F(Γ ′ ) = F(Γ), and the elliptic elements γ ∈ Γ ′ produce the indicated holomorphic sections s as the maps where z 0 is a fixed point of γ in the half-plane U .These sections are called the Weierstrass sections (in view of their connection with the Weierstrass points of hyperelliptic surface U/Γ).We describe these sections following [5].
Consider also the punctured fiber space F 0 (Γ) to be the largest open dense subset of F(Γ) on which Γ acts freely, and let V ′ (Γ) = F 0 (Γ)/Γ.For Γ with no elliptic elements, the universal covering space for of V ′ (g, n) = V ′ (Γ) is T(g, n + 1).
If Γ contains elliptic elements γ, then any holomorphic section T(Γ) → F(Γ) is determined by the maps (6) so that w µ (z 0 ) is exactly one fixed point of the corresponding map (4) in the fiber w µ (U ).These holomorphic sections take their values in the set V(Γ) \ V ′ (Γ) and do not provide in general sections of projection (5).
In the case of spaces T(1, 2) and T(2, 0), either of the corresponding curves V(1, 2) and V(2, 0) has a unique biholomorphic self-map γ of order two which maps each fiber into itself.The fixed-point locus of γ is a finite set of connected closed complex submanifolds of V ′ (g, n), and the restriction of the map (5) to one of these submanifolds is holomorphic map onto T(0, n); its inverse is a Weierstrass section.The restriction of γ to each fiber is a conformal involution of the corresponding hyperelliptic Riemann surface interchanging its sheets, and the fixed points of γ are the Weierstrass points on this surface.
As a special case of the Hubbard-Earle-Kra theorem [5], [9], we have: The curve V(0, 4) has for any its point x a unique holomorphic section s with s(π ′ and V(2, 0) ′ have holomorphic sections, which are their Weierstrass sections.
In particular, the curve V(2, 0) has six disjoint holomorphic sections corresponding to the Weierstrass points of hyperelliptic surfaces of genus two.

HOLOMORPHIC MAPS OF T(0, n) INTO UNIVERSAL TEICHM ÜLLER SPACE
3.1.The universal Teichmüller space T = Teich(U ) is the space of quasisymmetric homeomorphisms of the unit circle factorized by Möbius maps; all Teichmüller spaces have their isometric copies in T.
The canonical complex Banach structure on T is defined by factorization of the ball of the Beltrami coefficients (i.e., supported in the upper-half plane), letting µ 1 , µ 2 ∈ Belt(U ) 1 be equivalent if the corresponding quasiconformal maps w µ 1 , w µ 2 coincide on R = R ∪ {∞} = ∂U * (hence, on U * ).Such µ and the corresponding maps w µ are called T-equivalent.The equivalence classes [w µ ] T are in one-to-one correspondence with the Schwarzian derivatives S w in U * , which fill a bounded domain in the space B = B(U * ) (see 2.1).
The map φ T : µ → S w µ is holomorphic and descends to a biholomorphic map of the space T onto this domain, which we will identify with T. As was mentioned above, it contains as complex submanifolds the Teichmüller spaces of all hyperbolic Riemann surfaces and of Fuchsian groups.
We also define on this ball another equivalence relation, letting µ, ν ∈ Belt(U ) 1 be equivalent if w µ (a 0 j ) = w ν (a 0 j ) for all j and the homeomorphisms w µ , w ν are homotopic on the punctured sphere X a 0 .Let us call such µ and ν strongly n-equivalent.The proof of this lemma is given in [6].In view of Lemmas 1 and 4, the above factorizations of the ball Belt(U ) 1 generate (by descending to the equivalence classes) a holomorphic map χ of the underlying space T into T(0, n) = T(X a 0 ).This map is a split immersion (has local holomorphic sections), which is a consequence, for example, of the following existence theorem from [11], which we present here as Lemma 5. Let D be a finitely connected domain on the Riemann sphere C. Assume that there are a set E of positive two-dimensional Lebesgue measure and a finite number of points z 1 , ..., z m distinguished in D. Let α 1 , ..., α m be non-negative integers assigned to z 1 , ..., z m , respectively, so that α j = 0 if z j ∈ E.

3.2.
In fact, we have more, given by the following theorem which corrects the assertion of Lemma 3 in [12], as was mentioned in Preamble.
Theorem 2. The map χ is surjective and generates an open holomorphic map s of the space T(0, n) = T(X a 0 ) into the universal Teichmüller space T embedding T(0, n) into T as a (n − 3)dimensional submanifold.
Proof.The surjectivity of χ is a consequence of Lemma 2. To construct s, take a dense subset accumulating to all points of R and consider the punctured spheres X m a 0 = X a 0 \ {x 1 , . . ., x m } with m > 1.The equivalence relations on Belt(C) 1 for X m a 0 and X a 0 generate the corresponding holomorphic map χ m : T(X m a 0 ) → T(X a 0 ).Uniformizing the surfaces X a 0 and X m a 0 by the corresponding torsion free Fuchsian groups Γ 0 and Γ m 0 of the first kind acting discontinuously on U ∪ U * and applying to U * /Γ 0 and U * /Γ m 0 the construction from Section 2.3 (forgetting the additional punctures), one obtains similar to (3) the norm preserving isomorphism j m, * : which projects to the surfaces X a 0 and X m a 0 as the inclusion of the space Q(X a 0 ) of quadratic differentials corresponding to B(Γ 0 ) into the space Q(X m a 0 ), and (since the projection η m has local holomorphic sections) geometrically this relation yields a holomorphic embedding of the space T(Γ 0 ) into T(Γ m 0 ) as an (n − 3)-dimensional submanifold.Denote this embedding by s m .To investigate the limit function for m → ∞, we compose the maps s m with the canonical biholomorphic isomorphisms and this is a collection of the Schwarzians S f m (z) corresponding to the points X a of T(X a 0 ).So for any surface X a , we have s m (X a ) = S f m (z).( 7) Each Γ m 0 is the covering group of the universal cover h m : U * → X a m 0 , which can be normalized (conjugating appropriately Γ m 0 ) by h m (−i) = −i, h ′ m (−i) > 0. Take its fundamental polygon P m obtained as the union of the circular m-gon in U * centered at z = −i with the zero angles at the vertices and its reflection with respect to one of the boundary arcs.These polygons increasingly exhaust the half-plane U * from inside; hence, by the Carathéodory kernel theorem, the maps h m converge to the identity map locally uniformly in U * .
Since the set of punctures e is dense on R, it completely determines the equivalence classes [w µ ] and S w µ as the points of the universal space T. The limit function s = lim m→∞ s m maps T(X a 0 ) = T(0, n) into the space T and also is canonically defined by the marked spheres X a .
Similar to (7), the function s is represented as the Schwarzian of some univalent function f n on U * with quasiconformal extension to C determined by X a .Then, by the well-known property of elements in the functional spaces with sup norms, s is holomorphic also in B-norm on T.
Lemma 5 yields that s is a locally open map from T(X a 0 ) to T. So the image s(T(X a 0 )) is an (n − 3)-dimensional complex submanifold in T biholomorphically equivalent to T(Γ 0 ).The proof of Theorem 2 is completed.
The holomorphy property indicated above is based on the following lemma of Earle [4].Lemma 6.Let E, T be open subsets of complex Banach spaces X, Y and B(E) be a Banach space of holomorphic functions on E with sup norm.If ϕ(x, t) is a bounded map E × T → B(E) such that t → ϕ(x, t) is holomorphic for each x ∈ E, then the map ϕ is holomorphic.
Holomorphy of ϕ(x, t) in t for fixed x implies the existence of complex directional derivatives while the boundedness of ϕ in sup norm provides the uniform estimate , for sufficiently small |c| and v Y .
3.2.Now the desired holomorphic homotopy of T(0, n) = T(X a 0 ) into its base point is constructed as follows.
Using the canonical embedding of T(0, n) in T via T(Γ 0 ), we define on the space T(Γ 0 ) a holomorphic homotopy applying the maps This point-wise equality determines holomorphic map η(ϕ, t) = S w µ t : T × D → T with η(0, t) = 0, η(ϕ, 0) = 0, η(ϕ, 1) = ϕ.It is not compatible with the group Γ 0 ; hence, there are images η(ϕ, t) = S w µ t which are located in T outside of T(Γ 0 ).Composition of η(ϕ, t) with maps χ and s carries these images to the points of the space T(0, n) = T(X a 0 ) and implies the desired holomorphic homotopy of T(0, n).
Since T(0, 4) is (up to a biholomorphic equivalence) a simply connected bounded Jordan domain D ⊂ C containing the origin, there is a holomorphic isotopy h(ζ, t) : D × [0, 1] → D with h(ζ, 0) = ζ, h(z, 1) = 0. Using this isotopy, we define a homotopy h 1 (ϕ, t) on T(0, 5), which carries each point x = (S w µ , w µ ( a 0 2 )) ∈ T(0, 5) to its image on the section s passing from x, that is where a 2 is the common point of the fiber h(ϕ) and the selected section s.The holomorphy of this homotopy in variables x = (S w µ , w µ ( a 0 2 )) for any fixed t ∈ [0, 1] follows from Lemmas 1, 2 and the Bers isomorphism theorem.The limit map carries each fiber w µ (U ), to the initial half-plane U .
There is a canonical holomorphic isotopy of U into its point corresponding to the origin of T(0, 5).Now put h(x, t) to be equal to h 1 (x, 2t) for t ≤ 1/2 and equal to h 2 (x, 2t − 1) for x = ζ ∈ U and 1/2 ≤ t ≤ 1.This function is holomorphic in x ∈ T(0, 5) for any fixed t ∈ [0, 1] and hence provides the desired holomorphic homotopy of the space T(0, 5) into its base point.
We prescribe to each ordered sextuple a = {0, 1, a 1 , a 2 , a 3 , ∞} of distinct points the corresponding punctured sphere and the two-sheeted closed hyperelliptic surface X a of genus two with the branch points 0, 1, a 1 , a 2 , a 3 , ∞.The corresponding Teichmüller spaces T(0, 6) and T(2, 0) coincide up to a natural biholomorphic isomorphism.Note also that the collections a = {0, 1, a 1 , a 2 , a 3 , ∞} provide the local complex coordinates on each from the spaces T(0, 6) and T(2, 0).In view of the symmetry of hyperelliptic surfaces, it suffices to deal with the Beltrami differentials µdz/dz on X a , which are compatible with conformal involution J a of X a , hence, satisfy µ(J a z) = µ(z)J ′ a /J ′ a .In other words, these µ are obtained by lifting to X a of the Beltrami coefficients on X a .This extends Lemma 2 and its consequences on holomorphy in the neighborhoods of the boundary interpolation points to the corresponding two-sheeted disks on hyperelliptic surfaces.Fix a base point of T(2, 0) determining a Fuchsian group Γ for which T(Γ) = T(2, 0).The corresponding Teichmüller curve V(2, 0) is a 4-dimensional complex analytic manifold with projection π 1 : V(2, 0) → T(2, 0) onto T(2, 0) such that for every ϕ ∈ T(2, 0) the fiber π −1 1 (ϕ) is a hyperelliptic surface, determined by ϕ (see 2.4).
Due to assertion (b) of Lemma 3, this curve has for any point X a = (S w µ 1 , w µ 1 ( a 0 n−3 )) ∈ T(2, 0)) six distinct holomorphic sections s 1 , . . ., s 6 , corresponding to the Weierstrass points of the surface X a , with s j (π 1 (X a )) = X a , and either from these sections has one common point with every fiber over T(2, 0).We lift these sections to the Bers fiber space F(Γ) covering V(2, 0).
As was mentioned in 2.4, these sections are generated by the space F(Γ ′ ) = F(Γ) corresponding to the extension Γ ′ of Γ, for which Γ is a subgroup of index two.Every section s j acts on T(Γ ′ ) via (6), where z 0 is now the corresponding Weierstrass point of hyperelliptic surface X a , and s j is compatible with the action (2) of the Bers isomorphism.
Thus each s j descends to a holomorphic map s j : T(0, 6) → V(0, 6) of the underlying space T(0, 6) for the punctured spheres (10).We choose one from these maps and denote it by s.
The features of sections s j provide that the descended map s also determines for each point z 0 ∈ X a its unique image on every fiber w µ (X a ) with µ ∈ Belt(X a ) 1 , and this image is the point w µ (z 0 ).
The homotopy ( 12) is well defined on T(0, 6) × [0, 1] and contracts the set T(0, 6) into the fiber U over the base point.It is holomorphic with respect to the space variable x = (j * ϕ, z) for any fixed t ∈ [0, 1] and continuous in both variables.
It remains to combine this homotopy h 1 with the additional homotopy (9) of U into its point corresponding to the origin of T(0, 6).This provides the desired homotopy h and completes the proof of Theorem 1.