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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 $$(\mathbb{Z} \times B \Gamma_\infty^+)_p^\wedge \simeq\Omega^\infty (E_0) \times\Omega^\infty (E_1) \dots \times\Omega^\infty(E_{p-3}) \times W_p$$.

##### MSC:
 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)
##### Keywords:
transfer map; Mumford’s conjecture; splitting theorem
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