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In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin ( 2 ) \operatorname {Pin}(2) -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the Pin ( 2 ) \operatorname {Pin}(2) -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from B Pin ( 2 ) B\operatorname {Pin}(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j j -based Atiyah–Hirzebruch spectral sequence.
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We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $$\#^{g}S^{n}\times S^{n}$$ relative to a disc in a stable range, for $$2n\geqslant 6$$ . Our calculation is also valid for $2n=2$ assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.more » « less
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Abstract The Torelli group of $$W_g = \#^g S^n \times S^n$$ is the group of diffeomorphisms of $$W_g$$ fixing a disc that act trivially on $$H_n(W_g;\mathbb{Z} )$$ . The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $$\text{Sp}_{2g}(\mathbb{Z} )$$ or $$\text{O}_{g,g}(\mathbb{Z} )$$ . In this article we prove that for $$2n \geq 6$$ and $$g \geq 2$$ , they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent.more » « less
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We prove a new kind of stabilisation result, “secondary homological stability,” for the homology of mapping class groups of orientable surfaces with one boundary component. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of E2-algebras, which have no E2-cells below a certain vanishing line.more » « less
