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Register a new userCycles, submanifolds, and structures on normal bundles
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We give explicit examples of degree 3 cohomology classes not Poincare dual to submanifolds, and discuss the realisability of homology classes by submanifolds with Spin-C normal bundles.
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Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmuller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchins equations. Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to construct geometric structures in some interesting cases. In this paper, we will explain the main ideas behind this theory and we will survey some recent results in this direction, which are joint work with Qiongling Li.
We discuss a model for associative submanifolds in $G_{2}$-manifolds with K3 fibrations, in the adiabatic limit. The model involves graphs in a 3-manifold whose edges are locally gradient flow lines. We show that this model produces analogues of known singularity formation phenomena for associative submanifolds.
We define the secondary invariants L^2- eta and -rho forms for families of generalized Dirac operators on normal coverings of fibre bundles. On the covering family we assume transversally smooth spectral projections, and Novikov--Shubin invariants bigger than 3(dim B+1) to treat the large time asymptotic for general operators. In the particular case of a bundle of spin manifolds, we study the L^2- rho class in relation to the space of positive scalar curvature vertical metrics.
We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give bounds that are super-exponential in each of three variables: number of punctures, number of boundary components, and genus, generalizing work of Fullarton-Putman. Along the way, we give a simplified account of a theorem of Harer explaining how to relate the homotopy type of the curve complex of a multiply-punctured surface to the curve complex of a once-punctured surface through a process that can be viewed as an analogue of a Birman exact sequence for curve complexes. As an application, we prove upper and lower bounds on the coherent cohomological dimension of the moduli space of curves with marked points. For $g leq 5$, we compute this coherent cohomological dimension for any number of marked points. In contrast to our bounds on cohomology, when the surface has $n geq1$ marked points, these bounds turn out to be independent of $n$, and depend only on the genus.
We prove a homological characterization of $Q$-manifolds bundles over $C$-spaces. This provides a partial answer to Question QM22 from cite{w}.
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