Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 23–30

A NOTE OF THE TORELLI SPACES OF NON-ORIENTABLE COMPACT KLEIN SURFACES

Pablo Arés Gastesi

Tata Institute, School of Mathematics Bombay 400 005, India; pablo.pablo@math.tifr.res.in

Abstract. The Torelli group of a compact surface consists of the homotopy classes of homeomorphisms that act like the identity on the first homology group of the surface. In this note we show that the Torelli group of a non-orientable compact surface acts without fixed points on the Teichmüller space of the surface.

1. Statement of results

The purpose of this paper is to extend the construction of Torelli spaces of compact Riemann surfaces to the case of compact non-orientable surfaces. We will show, by topological methods, that if a diffeomorphism of a Klein surface acts trivially in the first homology group, then it is homotopic to the identity. This allows us to define the Torelli spaces of Klein surfaces, and show that they are smooth real submanifolds of the Torelli spaces of compact Riemann surfaces.

We give a more precise statement of our results. Let Σ be a smooth compact non-orientable surface. The Teichmüller space T (Σ) of Σ is defined as T (Σ) = ℳ (Σ)/ Diff_{0}(Σ), where ℳ (Σ) is the set of Klein surface structures on Σ that agree with the given smooth structure, and Diff_{0}(Σ) is the group of diffeomorphisms of Σ homotopic to the identity [7, p. 145]. The modular or mapping class group, Mod(Σ) = Diff(Σ)/ Diff_{0}(Σ), acts on T (Σ) by pull-back of structures. We define the Torelli group U(Σ) as the subgroup of Mod(Σ) consisting of the mapping classes that act like the identity on H_{1}(Σ, Z). The parallel result to the following theorem is a classical fact in Riemann surfaces theory.

Theorem 3.2. Let Σ be a smooth compact non-orientable surface of arithmetic genus g ≥ 2. Let f ∈ U(Σ), and suppose that there exists a Klein surface structure S on Σ such that f: (Σ, S) → (Σ, S) is dianalytic. Then f = id. Therefore, the Torelli group U(Σ) acts fixed-points free on T (Σ), and the Torelli space Tor(Σ) = T (Σ)/U(Σ) is a smooth real manifold of dimension 3g − 3.

Given a non-orientable surface Σ with a Klein surface structure S, there exists an unramified double cover (Σ^{c}, X), where Σ^{c} is an orientable surface of genus g, and X is a Riemann surface structure on Σ^{c}. The surface Σ^{c} has an involution σ, which is anti-holomorphic on the structure X, such that Σ is isomorphic to Σ^{c}/⟨σ⟩. The mapping σ induces in a natural way involutions σ^{∗} and σ̃ on the Teichmüller space T (Σ^{c}) and the Torelli space T (Σ^{c}), respectively. It is a classical result that T (Σ) can be identified with the set of fixed points of σ^{∗} on T (Σ^{c}).

1991 Mathematics Subject Classification: Primary 30F50; Secondary 14H40.

Proposition 3.4. The Torelli space Tor(Σ) can be identified with the set of fixed points of σ̃ on Tor(Σ^{c}).

2. Some general facts about Riemann and Klein surfaces

A Klein surface (or dianalytic) structure S on a surface without boundary Σ is given by a covering of Σ by open sets {U_{i}}_{i}, and a collection of homeomorphisms z_{i}: U_{i} → V_{i}, where V_{i} are open subsets of C, such that if U_{i} ∩ U_{j} ≠ ∅, then the mapping z_{i} ∘ z_{j}^{−1} is holomorphic or anti-holomorphic (conjugate of a holomorphic function) [1]. Klein surfaces are the generalisation of Riemann surfaces to the non-orientable case. We will write (Σ, S) or S for a Klein surface, depending on the context.

A non-orientable surface Σ is homeomorphic to the connected sum of p ≥ 1 real projective planes [2]. The integer p is called the topological genus of Σ. However, we will use the arithmetic genus, which is defined as g = p − 1 [7]. If g = 2n, the fundamental group of Σ has a presentation given by generators c, a_{1}, …, a_{n}, b_{1}, …, b_{n}, satisfying the relation c^{2} ∏_{j=1}^{n}[a_{j}, b_{j}] = 1, where [a, b] = aba^{−1}b^{−1} . If the genus of Σ is odd, g = 2n + 1, we can choose generators of the fundamental group c, d, a_{1}, …, a_{n}, …, b_{1}, …, b_{n}, that satisfy c^{2}d^{2} ∏_{j=1}^{n}[a_{j}, b_{j}] = 1.

Throughout this paper, all surfaces are assumed to be compact without boundary, of genus g ≥ 2.

Given a compact non-orientable surface Σ of genus g, there exists a compact orientable surface Σ^{c}, of genus g, and a double covering map: π: Σ^{c} → Σ [2]. If Σ has a Klein surface structure S, then it is possible to give a Riemann surface structure to X. Moreover, there exist local coordinates z and w, on Σ^{c} and Σ, respectively, such that the mapping w ∘ π ∘ z^{−1} is holomorphic. The pair (Σ^{c}, X) together with π is called the complex double [1] of (Σ, S). By an abuse of notation, we will refer to Σ^{c} as the complex double, when we are only interested in the topological aspects.

Let Y denote a compact orientable smooth surface, with a fixed orientation. The Teichmüller space T (Y) is defined as T (Y) = ℳ (Y)/ Diff_{0}(Y). Here ℳ (Y) is the set of Riemann surface structures on M that agree with the given orientation and smooth structure [7]. The group Diff_{0}(Y) consists of the diffeomorphisms of Y that are homotopic to the identity. The modular group Mod(Y) = Diff^{+}(Y)/ Diff_{0}(Y), consisting of homotopy classes of orientation preserving diffeomorphisms of Y , acts on T (Y) by pull-back: if [f] ∈ Mod(Y) and [X] ∈ T (Y), then [f]^{∗}([X]) is defined as the class [f^{∗}(X)] of the Riemann surface structure that makes f: (Y, f^{∗}(X)) → (Y, X) biholomorphic. It is a well-known result that Mod(Y) acts on T (Y) with fixed points, corresponding to surfaces with automorphisms. A subgroup G of Mod(Y) is said to have the Hurwitz–Serre property [5] if for any element [g] ∈ G, such that there exists a point [X] in T (Y) for which g: (Y, X) → (Y, X) is biholomorphic, we have that [g] = [id]. A group with this property acts fixed-points free on T (Y). The Torelli group U(Y) consists of those elements of Mod(Y) that act trivially on H_{1}(Y, Z); it is a classical fact that U(Y) satisfies the Hurwitz–Serre property [3]. The quotient space Tor(Y) := T (Y)/U(Y) is called the Torelli space of Y .

Teichmüller spaces and modular groups of non-orientable surfaces are defined in a similar way, removing all the conditions that involve the orientability of the surface, and substituting Riemann surface structures by Klein surfaces.

3. Torelli groups of non-orientable surfaces

In this section, we first prove that the Torelli groups of compact non-orientable surfaces have the Hurwitz–Serre property. We do this by lifting homeomorphisms from the surface to its complex double, and then using the fact that the Torelli groups of Riemann surfaces satisfy the above mentioned property. We use this fact to show that the Torelli spaces of Klein surfaces are smooth manifolds, which can be embedded into the Torelli spaces of the complex doubles, in a situation similar to what happens between Teichmüller spaces.

Theorem 3.1. Let Σ be a compact non-orientable surface, and let f be a homeomorphism of Σ. Assume that f acts trivially on H_{1}(Σ, Z). Let f̃ be the unique orientation lift of f to the complex double Σ^{c}. Then f̃ acts trivially on H_{1}(Σ^{c}, Z).

Proof. Our proof will be divided in two cases, depending on whether the genus of Σ is even or odd; we will provide full details in the first case, and only a sketch of the second situation.

Let f: Σ → Σ be a homeomorphism, and let f̃ the the unique orientation preserving lift of f to the complex double, so that the following diagram is commutative [6]:

\xymatrix{Σ^{c} \ar[r]^{f̃} \ar[d]_{π} & Σ^{c} \ar[d]^{π} \\ Σ \ar[r]^{f} & Σ.}

Let f_{#} and f̃_{#} denote the corresponding mappings induced by f and f̃ on the first homology groups of Σ and Σ^{c} respectively. By hypothesis we have that f_{#} = id, and we want to show that f̃_{#} = id.

We start by recalling how the surface Σ^{c} is constructed, from a topological point of view; the reader can find more details in [2]. Assume that Σ has genus g = 2n. By the presentation of the fundamental group of Σ given in Section 2, we can view this surface as a (2n + 2)-polygon, with the sides identified by the relation of the fundamental group. The surface Σ^{c} is given by two polygons, with boundary relations:

c_{1}c_{2} ∏_{j=1}^{n}[a_{j,1}, b_{j,1}] = 1  and  c_{2}c_{1} ∏_{j=1}^{n}[a_{j,2}, b_{j,2}] = 1.

To obtain a single relation, we find the value of c_{2} on the right-hand side equation and substitute it on the left-hand side one (equivalently, we glue the polygons by the c_{2} sides):

c_{2} = (∏_{j=1}^{n}[b_{n+1−j,2}, a_{n+1−j,2}])c_{1}^{−1};

so

\begin{matrix}c_{1}(∏_{j=1}^{n} & [b_{n+1−j,2}, a_{n+1−j,2}])c_{1}^{−1}(∏_{j=1}^{n}[a_{j,1}, b_{j,2}]) \\ = (∏_{j=1}^{n}[c_{1}b_{n+1−j,2}c_{1}^{−1}, c_{1}a_{n+1−j,2}c_{1}^{−1}])(∏_{j=1}^{n}[a_{j,1}, b_{j,2}]) = 1. \end{matrix}

The above expression shows that Σ^{c} is a compact surface of genus g. We can take the following paths as generators of the fundamental group of Σ^{c}:

\begin{aligned}α_{1} & = c_{1}b_{n,2}c_{1}^{−1}, …, α_{n} = c_{1}b_{1,2}c_{1}^{−1}, α_{n+1} = a_{1,1}, …, α_{2n} = a_{n,1}, \\ β_{1} & = c_{1}a_{n,2}c_{1}^{−1}, …, β_{n} = c_{1}a_{1,2}c_{1}^{−1}, β_{n+1} = b_{1,1}, …, β_{2n} = b_{n,1}. \end{aligned}

These loops satisfy ∏_{j=1}^{n}[α_{j}, β_{j}] = 1. Let ℬ and ℬ^{c} denote the basis in homology induced by the above two sets of generators of the fundamental groups of Σ and Σ^{c}, respectively. By an abuse of notation, we will use the same letters for paths of the fundamental groups and the corresponding classes in homology. It is not difficult to see that ℬ^{c} is a symplectic basis of H_{1}(Σ^{c}, Z); that is, its intersection matrix is

J = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) ,

where I is the identity matrix of order g. The covering map π has the following associated matrix for its action on homology, with respect to the two bases given above, ordered as ℬ = {{c}, {a_{1}, …, a_{n}}, {b_{1}, …, b_{n}}} and ℬ^{c} = {{α_{1}, …, α_{n}}, {α_{n+1}, …, α_{2n}}, {β_{1}, …, β_{n}}, {β_{n+1}, …, β_{2n}}}:

π_{#} = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & I & K & 0 \\ K & 0 & 0 & I \end{matrix} )

where K is the matrix

K = ( \begin{matrix}0 & … & … & 1 \\ 0 & … & 1 & 0 \\ ⋮ & & & ⋮ \\ 1 & 0 & … & 0 \end{matrix} ) .

The symmetry σ maps a_{j,1} (respectively b_{j,1} ) to a_{j,2} (respectively b_{j,2} ); therefore, its action on H_{1}(Σ^{c}, Z) is given by the matrix K (of order 2g). Let f̃_{#} = (A_{ij})_{i,j=1}^{4} be the matrix associated to the mapping f̃. Then f̃_{#} must satisfy

(3.1)  (a) π_{#}f̃_{#} = π_{#},  (b) f̃_{#}σ_{#} = σ_{#}f̃_{#},  (c) f̃_{#}^{t} Jf̃_{#} = J.

Equation (a) of (3.1) is due to the fact that f acts trivially on H_{1}(Σ, Z). The uniqueness of the orientation-preserving lift gives us (b) above. In (c), which is true for any orientation-preserving homeomorphism of Σ^{c} [4, Theorem N13, p. 178], f̃_{#}^{t} denotes the transpose matrix. Equation (3.1a) is equivalent to the following:

\{ \begin{matrix}A_{21} + KA_{31} & = 0 \\ A_{22} + KA_{32} & = I \\ A_{23} + KA_{33} & = K \\ A_{24} + KA_{34} & = 0 \end{matrix}  \begin{aligned}KA_{11} + A_{41} & = K \\ KA_{12} + A_{42} & = 0 \\ KA_{13} + A_{43} & = 0 \\ KA_{14} + A_{44} & = I. \end{aligned}

Therefore, we have that the matrix f̃_{#} is given by

f̃_{#} = ( \begin{matrix}A_{11} & A_{12} & A_{13} & A_{14} \\ A_{21} & A_{22} & A_{23} & A_{24} \\ −KA_{21} & K − KA_{22} & I − KA_{23} & −KA_{24} \\ K − KA_{11} & −KA_{12} & −KA_{13} & I − KA_{14} \end{matrix} ) .

From (3.1)(b) we obtain:

(3.2)  \{ \begin{matrix}A_{14}K & = K(K − KA_{11}) \\ A_{13}K & = K(−KA_{12}) \\ A_{12}K & = K(KA_{13}) \\ A_{11}K & = K(I − KA_{14}) \end{matrix}  \begin{aligned}A_{24}K & = K(−KA_{21}) \\ A_{23}K & = K(K − KA_{22}) \\ A_{22}K & = K(I − KA_{23}) \\ A_{21}K & = K(−KA_{24}). \end{aligned}

Looking at the first row of the equation (c) of (3.1) we get the following identities, after using (3.2) to simplify the result:

\{ \begin{matrix}A_{21}^{t}K − KA_{21} & = 0 \\ A_{11}^{t}K − KA_{22} & = 0 \\ −2I + A_{11}^{t} + KA_{22}K & = 0 \end{matrix}

Solving these equations we obtain A_{21} = 0 and A_{11} = I, which imply A_{24} = A_{14} = 0. By a similar set of equations given by the second row of the relation (c) of (3.1) we get A_{22} = I and A_{12} = 0. By (3.2) we have A_{23} = A_{13} = 0, which shows that f̃_{#} is the identity matrix.

For the case of odd genus g = 2n + 1, we will use the presentation of the fundamental group of Σ given in Section 2. The complex double Σ^{c} is homeomorphic to two polygons with the following boundary relations:

c_{1}c_{2}d_{1}d_{2} ∏_{j=1}^{n}[a_{j,1}, b_{j,2}] = 1  and  c_{2}c_{1}d_{2}d_{1} ∏_{j=1}^{n}[a_{j,2}, b_{j,1}] = 1.

Calculating as in the previous situation, we have

d_{2} = c_{1}^{−1}c_{2}^{−1}(∏_{j=1}^{n}[b_{n+1−j,2}, a_{n+1−j,2}])d_{1}^{−1},

which can be substituted into the first boundary equation of Σ^{c} to get

\begin{matrix}c_{1}c_{2}c_{1}^{−1}c_{2}^{−1}(∏_{j=1}^{n}[b_{n+1−j,2}, a_{n+1−j,2}])(∏_{j=1}^{n}[a_{j,1}, b_{j,1}]) \\ = c_{1}c_{2}d_{1}c_{1}^{−1}c_{2}^{−1}d_{1}^{−1}(∏_{j=1}^{n}[d_{1}b_{n+1−j,2}d_{1}^{−1}, d_{1}a_{n+1−j,2}d_{1}^{−1}])(∏_{j=1}^{n}[a_{j,1}, b_{j,1}]) = 1. \end{matrix}

We therefore obtain the following set of generators of the fundamental group of Σ^{c}:

α_{1} = c_{1}d_{1}^{−1}, α_{2} = d_{1}b_{n,2}d_{1}^{−1}, …, α_{n+1} = d_{1}b_{1,2}d_{1}^{−1}, α_{n+2} = a_{1,1}, …, α_{2n+1} = a_{n,1},

β_{1} = d_{1}c_{2}, β_{2} = d_{1}a_{n,2}d_{1}^{−1}, …, β_{n+1} = d_{1}a_{1,2}d_{1}^{−1}, β_{n+2} = b_{1,1}, …, β_{2n+1} = b_{n,1},

which satisfy the relation ∏_{j=1}^{2n+1}[α_{j}, β_{j}] = 1. Although the basis {α_{1}, …, α_{2n+1} , β_{1}, …, β_{2n+1}} is symplectic, computations are easier if we arrange the generators in the following way ℬ^{c} = {α_{1}, β_{1}, α_{2}, …, α_{2n+1}, β_{2}, …, β_{2n+1}}, whose intersection matrix is:

( \begin{matrix}N & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & I \\ 0 & 0 & −I & 0 & 0 \\ 0 & −I & 0 & 0 & 0 \end{matrix} ) .

Here

N = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .

With respect to the basis ℬ = {{c, d}, {a_{1}, …, a_{n}}, {b_{1}, …, b_{n}}} and ℬ^{c} = {{α_{1}, β_{1}}, {α_{2}, …, α_{n+1}}, {α_{n+2}, …, α_{2n+1}}, {β_{2}, …, β_{n+1}}, {β_{n+2}, …, β_{2n+1}}}, we have that the action of σ on the first homology group is given by

σ_{#} = ( \begin{matrix}M & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & O & K \\ 0 & 0 & 0 & K & 0 \\ 0 & 0 & K & 0 & 0 \\ 0 & K & 0 & 0 & 0 \end{matrix} ) ,

where

M = ( \begin{matrix}1 & 0 \\ 2 & −1 \end{matrix} ) .

To see how M is obtained, observe that in homology we have α_{1} = c_{1} − d_{1} and β_{1} = d_{1} + c_{2} . Therefore, σ(α_{1}) = c_{2} − d_{2} and σ(β_{1}) = d_{2} + c_{1} . Substituting the value of d_{2} obtained previously we get the matrix M. Similarly, we have that the covering map π: Σ^{c} → Σ has the following associated matrix for its action on homology:

π_{#} = ( \begin{matrix}L & 0 & 0 & 0 & 0 \\ 0 & 0 & I & K & 0 \\ 0 & K & 0 & 0 & I \end{matrix} ) ,

where

L = ( \begin{matrix}1 & −1 \\ 1 & 1 \end{matrix} ) .

The rest of the proof is similar to the even genus case.

A mapping f: (Σ_{1}, S_{1}) → (Σ_{2}, S_{2}) between Klein surfaces is called dianalytic if there exist local coordinates z_{j} in Σ_{j}, j = 1, 2, such that the mapping z_{2}∘f ∘z_{1}^{−1} is holomorphic or anti-holomorphic. The above computations show that the Torelli group U(Σ) of a compact non-orientable Klein surface satisfies the Hurwitz–Serre property.

Theorem 3.2. Let Σ be a compact non-orientable surface of genus g ≥ 2. Let [f] ∈ U(Σ), and suppose that there exists a Klein surface structure S on Σ such that f: (Σ, S) → (Σ, S) is dianalytic. Then f = id.

Corollary 3.3. The Torelli space Tor(Σ) = T (Σ)/U(Σ) is a smooth real manifold of dimension 3g − 3.

Proof of the theorem. Since f is dianalytic on the Klein surface (Σ, S), the orientation preserving lift f̃ is biholomorphic on the Riemann surface (Σ^{c}, X). By the previous result we have that f̃ acts like the identity on H_{1}(Σ^{c}, Z). But then, by the classical theory of Riemann surfaces [3], we have that f̃ = id_{Σ^{c}}, which implies that f = id_{Σ} .

30 Pablo Arés Gastesi

The involution σ induces in a natural way a symmetry σ^{∗} on the Teichmüller space T (Σ^{c}). It is clear that σ^{∗} descends to a symmetry σ̃ of Tor(Σ^{c}).

Proposition 3.4. The Torelli space Tor(Σ) can be identified with the set of fixed points of σ̃ in Tor(Σ^{c}).

Proof. The proof follows immediately from the definition of Torelli spaces. In fact, we have that two points S_{1} and S_{2} of ℳ (Σ) project to the same point in Tor(Σ) if and only if there exists a dianalytic mapping h: (Σ, S_{1}) → (Σ, S_{2}) such that h_{#}: H_{1}(Σ, Z) → H_{1}(Σ, Z) is the identity. The rest of the proof is similar to the fact that T (Σ) can be identified with the set of fixed points of σ^{∗} in T (Σ^{c}); see [7] for more details.

References

[1] Alling, N., and N. Greenleaf: Foundations of the Theory of Klein surfaces. - Lecture Notes in Math. 219, Springer-Verlag, Berlin–Heidelberg–New York, 1971.

[2] Blackett, D.W.: Elementary Topology, a Combinatorial and Algebraic Approach. - Texbooks in Math. 219, Academic Press, New York–London, 1967.

[3] Farkas, F., and I. Kra: Riemann Surfaces, 2nd ed. - Grad. Texts in Math. 72, Springer-Verlag, New York–Heidelberg–Berlin, 1992.

[4] Magnus, W., A. Karrass, and D. Solitar: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. - Pure Appl. Math. XIII, Interscience Publishers, New York–London–Sydney, 1966.

[5] Nag, S.: The Complex Analytic Theory of Teichmüller Spaces. - John Wiley & Sons, 1988.

[6] Seppälä, M.: Teichmüller spaces of Klein surfaces. - Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 15, 1978.

[7] Seppälä, M., and T. Sorvali: Geometry of Riemann Surfaces and Teichmüller Spaces. - North-Holland, Amsterdam, 1992.

Received 20 November 1996