Related papers: Kreck-Stolz invariants for quaternionic line bundl…
We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper…
We show that the Reshetikhin-Turaev-Walker invariant of 3-manifolds can be normalized to obtain an invariant of 4-dimensional thickenings of 2-complexes. Moreover when the underlying semisimple tortile category comes from the…
In an earlier paper (math.SG/0101206), we introduced Floer homology theories associated to closed, oriented three-manifolds Y and SpinC structures. In the present paper, we give calculations and study the properties of these invariants. The…
The equations of motion and the Bianchi identity of the C-field in M-theory are encoded in terms of the signature operator. We then reformulate the topological part of the action in M-theory using the signature, which leads to connections…
Kreck's modified surgery gives an approach to classifying smooth $2n$-manifolds up to stable diffeomorphism, i.e. up to connected sum with copies of $S^n \times S^n$. In dimension 4, we use a combination of modified and classical surgery to…
Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of…
In [BK], it is shown that the Turaev-Viro invariants defined for a spherical fusion category $\mathcal{A}$ extends to invariants of 3-manifolds with corners. In [Kir], an equivalent formulation for the 2-1 part of the theory (2-manifolds…
We show that the Atiyah-Patodi-Singer reduced $\eta$-invariant of the twisted Dirac operator on a closed $4m-1$ dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a…
We establish comparison results between the Hasse-Witt invariants w_t(E) of a symmetric bundle E over a scheme and the invariants of one of its twists E_{\alpha}. For general twists we describe the difference between w_t(E) and…
We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional…
We find lower bounds on the rank of a "real" vector bundle over an involutive space, such that "real" vector bundles of higher rank have a trivial summand and such that a stable isomorphism for such bundles implies ordinary isomorphism. We…
We use new homotopy-theoretic tools to prove the existence of smooth $U(1)$- and $Sp(1)$-actions on infinite families of exotic spheres. Such families of spheres are propagated by the complex and quaternionic analogues of the Mahowald…
In this book we prove unified classification results for equivariant principal bundles when the topological structure group is truncated. The conceptually transparent proof invokes a smooth Oka principle, which becomes available after…
We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we…
We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by C_8, which are complete on a complex line bundle over CP^3. The principal orbits are S^7, described…
In this note we describe the physics of equivalence of the Seiberg-Witten invariants of 4-manifolds and certain Gromov-Witten invariants defined by pseudo-holomorphic curves. We show that physics of the pseudo-holomorphic curves should be…
Let $\phi : S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $\phi$ to K has exactly two (non-degenerate) critical points (and K is…
With this paper we extend our studies [1] on polarized beams by distilling tools from the theory of principal bundles. Four major theorems are presented, one which ties invariant fields with the notion of normal form, one which allows one…
We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each…
Ozsv\'ath and Szab\'o used the knot filtration on $\widehat{CF}(S^3)$ to define the $\tau$-invariant for knots in the 3-sphere. In this article, we generalize their construction and define a collection of $\tau$-invariants associated to a…