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The stable mapping class group and \(Q(\mathbb C P^ \infty_ +)\). (English) Zbl 1050.55007
Let \(F\) be an oriented surface of genus \(g\), \(\text{Diff} (F; \partial)\) the topological group of orientation preserving diffeomorphisms fixing the (possibly empty) boundary. If \(g>1\), the components of \(\text{Diff} (F;\partial)\) are contractible and BDiff\((F;\partial)= B\Gamma (F)\) where \(\Gamma(F)= \pi_0(\text{Diff}(F;\partial))\) is the mapping class group. The Mumford conjecture is that the rational cohomology of \(B \Gamma(F)\) in dimensions low compared to \(g\) is a polynomial algebra over classes \(\kappa_i\) in \(H^{2i}(B\Gamma (F); \mathbb{Q})\). Let \(F_{g, 1+1}\) be a surface with two boundary components and mapping class group \(\Gamma_{g,1+1}\). \(F_{g,1+1}\) is included in \(F_{g+1,1+1}\) by gluing a torus with two boundary components onto \(F_{g,1+1}\). One obtains maps \(B\Gamma^+_{g, 1+1}\to B \Gamma^+_{g+1,1+1}\) and \(B\Gamma^+_{g,1+1} \to B \Gamma^+_{g,1+1}\) where + is Quillen’s plus construction. The homotopy direct limit of these maps is \(B\Gamma_\infty^+\). Mumford’s conjecture takes the form \(H^*(B \Gamma_\infty^+; \mathbb{Q})=\mathbb{Q} [\kappa_1,\kappa_2,\dots]\). It was previously shown that \(\mathbb{Z}\times B\Gamma_\infty^+\) has an infinite loop space structure. There is an infinite loop map \(\alpha_\infty:\mathbb{Z}\times B\Gamma_\infty^+ \to \Omega^\infty \mathbb{C} P_{-1}^\infty\) which is conjectured to be a homotopy equivalence. \(\Omega^\infty\mathbb{C} P_{-1}^\infty\) is the colim \(\Omega^{2s+2} Th(-L_s)\) and \(Th(-L_s)\) is the Thom space of the complementary \(\mathbb{C}^s\) bundle of the canonical line bundle over \(\mathbb{C} P^s\). This is an extension of Mumford’s conjecture. Pursuant to that conjecture the authors obtain the following splitting theorem. For \(p\) an odd prime, let the “Adams” splitting of \(\Sigma^\infty (\mathbb{C} P^\infty)_p^\wedge\) be \(E_0,E_1,\dots,E_{p-2}\). It is shown that there is an infinite loop space \(W_p\) such that \((\mathbb{Z} \times B \Gamma_\infty^+)_p^\wedge \simeq\Omega^\infty (E_0) \times\Omega^\infty (E_1) \dots \times\Omega^\infty(E_{p-3}) \times W_p\).

55P47 Infinite loop spaces
55R12 Transfer for fiber spaces and bundles in algebraic topology
57M99 General low-dimensional topology
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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