Related papers: The theta divisor and the Casson-Walker invariant
Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding we highlight a new area where mock modular forms start to play…
In this paper we further develop the theory of equivariant Seiberg-Witten-Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain the following applications. First, we show that the knot concordance…
We derive a formula for the $\bar\mu$-invariant of a Seifert fibered homology sphere in terms of the eta-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed…
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…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
In this paper, we introduce the concept of 3-alterfolds with embedded separating surfaces. When the separating surface is decorated by a spherical fusion category, we obtain quantum invariants of 3-alterfold, which is consistent with many…
We propose a general theory of the Open Gromov-Witten invariant on Calabi-Yau three-folds. We introduce the moduli space of multi-curves and show how it leads to invariants. Our construction is based on an idea of Witten. In the special…
We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an…
For any rational homology 3-sphere and one of its spin^{c}-structures, Ozsvath and Szabo defined a topological invariant, called d-invariant. Given a knot in the 3-sphere, the d-invariants associated with the prime-power-fold branched…
The eta invariant appears regularly in index theorems but is known to be directly computable from the spectrum only in certain examples of locally symmetric spaces of compact type. In this work, we derive some general formulas useful for…
In this paper, we study a model for $S^1$-equivariant monopole Floer homology for rational homology three-spheres via a homological device called $\mathcal{S}$-complex. Using the Chern-Simons-Dirac functional, we define an…
The t-invariant can be considered as the Turaev-Viro invariant of order 5 computed for integer colors only. We compute all values of the t-invariant for Seifert manifolds with base sphere and three singular fibers. As a result we show that…
We construct power series invariants of rational homology 3-spheres from quantum PSU(n)-invariants. The power series can be regarded as perturbative invariants corresponding to the contribution of the trivial connection in the hypothetical…
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose…
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…
A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This…
We define an integer valued invariant of homology spheres using the methods of SU(3) gauge theory and study its behavior under orientation reversal and connected sum.
In this paper we study isotopy classes of closed connected orientable surfaces in the standard $3$-sphere. Such a surface splits the $3$-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a…
We use Heegaard Floer homology with twisted coefficients to define numerical invariants for arbitrary closed 3-manifolds equipped torsion spin$^c$ structures, generalising the correction terms (or $d$--invariants) defined by Ozsv\'ath and…
We show that the Witten-Reshetikhin-Turaev SU(2) invariant and the Hennings invariant associated to the restricted quantum $sl_2$ are essentially the same for rational homology 3-spheres.