Friday, 19 June 2009

at.algebraic topology - Homotopy type of set of self homotopy-equivalences of a surface

Hi Andy,



Here is a proof for the case with marked points (see below for some ideas for the case of closed surfaces).



Proof: straight-line homotopy.



Less tersely: let HE0(Sigma,ast) be the identity component of the monoid of self-homotopy equivalences of Sigma fixing the basepoint; in particular, each finHE0(Sigma,ast) is homotopic rel ast to the identity. Fix a hyperbolic metric on Sigma and thus an identification of the universal cover widetildeSigma with the hyperbolic plane mathbbH2, and a basepoint ast in mathbbH2.



Each fcolonSigmatoSigma has a unique lift to mathbbH2 fixing the basepoint. because f acts trivially on pi1(Sigma), f commutes with the deck transformations. Thus we may take the straight-line homotopy ft(x)=tx(1t)f(x), where by this convex combination I mean to move with unit speed along the geodesic from f(x) to x. Since the deck transformations act by isometries on mathbbH2, this homotopy descends to Sigma; each ft is still a homotopy equivalence. We can perform this straight-line homotopy for all f simultaneously; since the lifts of f are uniformly continuous, this homotopy is continuous on HE0(Sigma,ast) and gives a contraction to the identity.



There must be some work needed to get from this to the case for closed surfaces, because this proof works for a genus 1 surface with marked point, and of course HE0(T2) is homotopy equivalent to T2 itself. But it seems to me like a reduction should be possible; I think the important thing is that pi1(Sigma) is centerless.



Acknowledgement: I learned this idea from Rita Jimenez Rolland, based on conversations she had with Mladen Bestvina about the related case of textAut(Fn).

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