Related papers: Dual 2-complexes in 4-manifolds

We prove a geometric refinement of Alexander duality for certain 2-complexes, the so-called gropes, embedded into 4-space. This refinement can be roughly formulated as saying that 4-dimensional Alexander duality preserves the disjoint Dwyer…

We identify the 2-groupoid of deformations of a gerbe on a smooth manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on infinite jets of smooth functions.

We present new explicit decompositions of manifolds via so-called fold maps into lower dimensional spaces. Fold maps form a nice class of so-called generic maps, generalizing Morse functions naturally. To understand the topologies and the…

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $\mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open…

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of…

For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology…

We show that for any smooth Hausdorff manifolds M and N, which are not necessarily second countable, paracompact or connected, any isomorphism from the algebra of smooth (real or complex) functions on N to the algebra of smooth functions on…

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if…

New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on…

We introduce the concept of a handlebody decomposition of a 3-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable 3-manifold are stably equivalent. As an…

We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.

We construct smooth manifolds with order two $\pi_1$ and even intersection forms which are irreducible, meaning they do not decompose into non-trivial connected sums. Their intersection forms being even implies that their universal covers…

We show that any finitely presented group with an index two subgroup is realized as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist.…

We introduce the notion of lef line bundles on a complex projective manifold. We prove that lef line bundles satisfy the Hard Lefschetz Theorem, the Lefschetz Decomposition and the Hodge-Riemann Bilinear Relations. We study proper…

We consider double plumbings of two disk bundles over spheres. We calculate the Heegaard-Floer homology with its absolute grading of the boundary of such a plumbing. Given a closed smooth 4-manifold $X$ and a suitable pair of classes in…

Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an…

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a…

Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4-manifolds with…

Let $W$ be a compact smooth $4$-manifold that deformation retract to a PL embedded closed surface. One can arrange the embedding to have at most one non-locally-flat point, and near the point the topology of the embedding is encoded in the…

Double forms are sections of the vector bundles $\Lambda^{k}T^*\mathcal{M}\otimes \Lambda^{m}T^*\mathcal{M}$, where in this work $(\mathcal{M},\mathfrak{g})$ is a compact Riemannian manifold with boundary. We study graded second-order…