The Spine Which Was No Spine

Let Tn be the Teichmüller space of flat metrics on the n-dimensional torus T and identify SLn Z with the corresponding mapping class group. We prove that the subset Y consisting of those points at which the systoles generate π1(T) is, for n ≥ 5, not contractible. In particular, Y is not a SLn Z-equivariant deformation retract of Tn. For n ≥ 2 let Tn be the Teichmüller space of flat metrics with unit volume on the n-dimensional torus T = R/Z. To be more precise, Tn is the set of equivalence classes of unit volume flat metrics on T where two metrics ρ and ρ′ are equivalent if there is an orientation preserving diffeomorphism φ ∈ Diff+(T) homotopic to the identity with ρ′ = φ∗ρ. We consider on the Teichmüller space Tn the topology with respect to which the classes of two flat metrics ρ and ρ′ are close if there is a diffeomorphism φ ∈ Diff+(T) homotopic to the identity such that ρ′ and φ∗ρ are close as tensors. Every element A ∈ SLn Z induces an orientation preserving diffeomorphism A ∈ Diff+(T) which is said to be linear. We obtain thus a right action of SLn Z on Tn: Tn × SLn Z → Tn, (ρ,A) 7→ A∗ρ which is properly discontinuous. There exists a finite index subgroup Γ of SLn Z which acts freely; in particular, the contractibility of Tn implies that for any such subgroup Γ, the quotient Tn/Γ is an EilenbergMacLane space for Γ. The systole syst(ρ) of a point ρ ∈ Tn is the length of the shortest homotopically essential geodesic in the flat torus (T, ρ). Let S(ρ) be the set of homotopy classes of geodesics in (T, ρ) with length syst(ρ); the elements in S(ρ) are known as the systoles of (T, ρ). Ash [1] proved that the systole function Tn → (0,∞), ρ 7→ syst(ρ) is a SLn Z-equivariant topological Morse function, and so it is not surprising that it can be used to construct a particularly nice SLn Zequivariant spine, i.e., deformation retract, of Tn. More precisely, the 1 2 ALEXANDRA PETTET & JUAN SOUTO following result was proved in a different language and much greater generality by Ash [2]: Theorem 1 (Ash). The subset X of Tn consisting of those points ρ with the property that S(ρ) generates a finite index subgroup of π1(T) is an SLn Z-equivariant spine of Tn. From a geometric point of view, that the systoles generate a finite index subgroup of π1(T) seems to be a peculiar condition. This led the authors to wonder whether the subset Y of Tn consisting of those points ρ ∈ Tn with the property that the systoles generate the full group π1(T) could be a SLn Z-equivariant deformation retract as well. For n = 2, 3 and 4, this is known, as for these cases the sets X and Y coincide [8, 9]. The goal of this note is to show that this fails to be true for n ≥ 5, although the complex Y is always a CW-complex of dimension n(n−1) 2 . Theorem 2. For n ≥ 5, the subset Y of Tn consisting of those points ρ with the property that S(ρ) generates π1(T) is not contractible and hence it is not a SLn Z-equivariant spine. Observe that Ash’s spine X , known as the well-rounded retract, is homeomorphic to a CW-complex with the same dimension as the virtual cohomological dimension vcdim(SLn Z) = n(n−1) 2 of SLn Z. The complex Y is also a CW-complex of the correct dimension. In order to prove Theorem 2, we make use of the well-known identification between the Teichmüller space Tn and the symmetric space Sn = SOn \ SLn R. We discuss this identification in Section 1. For the convenience of the reader, we also sketch briefly the proof of Theorem 1 in Section 2. Now let Γ be a torsion free finite index subgroup of SLn Z. The action of Γ on Sn is free and hence the quotient MΓ = Sn/Γ is a manifold. Borel and Serre [5] constructed a compact manifold M̄Γ with boundary ∂M̄Γ whose interior is homeomorphic to MΓ. In section 3 we briefly describe how to construct non-trivial homology classes in Hn(n−1) 2 (MΓ) and Hn−1(M̄Γ, ∂M̄Γ). These classes are then used in Section 4 to show that whenever Γ is as above and is contained in the kernel of the standard homomorphism SLn Z → SLn Z/2Z, the inclusion Y/Γ → MΓ is not surjective on the n(n−1) 2 -homology; Theorem 2 follows. We thank Martin Henk for showing us an example of a point X \Y . We also thank Mladen Bestvina for convincing us that there was no way that Y was a retract, and for almost completely proving it for us. The