Search

In this paper, we show that rational homology 3-spheres are ubiquitous from the viewpoint of Heegaard splitting. Let M = H+ UF H_ be a genus g Heegaa;rd splitting of a closed 3-manifold and c be a simple closed curve in F. Then there is a 3-manifold Mc which is obtained from M by horizontal Dehn surgery along c. We show that for c such that the homology class [c] is generic in the set of curve-represented homology classes Hs ⊂ H1(F), rank(H1(Mc,Q)) < max{ 1, rank(H1(M,Q)}. As a corollary, for a set of cusub>, c2, hellip;, cm}, m ≥ g, such that each [ci] is generic in Hs ⊂ H1(F), M(c1,c2,…,cm) is a rational homology 3-sphere. [ABSTRACT FROM AUTHOR]

This paper records two results which were inexplicably omitted from the paper on Pin structures on low dimensional manifolds in the LMS Lecture Note Series, volume 151, by Kirby and this author. Kirby declined to be listed as a coauthor of this paper. par A $Pin^{-}$-structure on a surface $X$ induces a quadratic enhancement of the mod $2$ intersection form, $qcolon H_1(X;mathbb {Z}/2mathbb {Z})to mathbb {Z}/4mathbb {Z}$. par Theorem 1.1 says that $q$ vanishes on the kernel of the map in homology to a bounding $3$-manifold. This is used by Kreck and Puppe in their paper in Homology, Homotopy and Applications, volume 10. The arXiv version, arXiv:0707.1599 [math.AT], referred to an email from the author to Kreck for the proof. A more polished and public proof seems desirable. par In Section 6 of the paper with Kirby, a $Pin^{-}$-structure is constructed on a surface $X$ dual to $w_2$ in an oriented 4-manifold, $M^4$. Theorem 2.1 says that $q$ vanishes on the Poincaré dual to the image of $H^1(M;mathbb {Z}/2mathbb {Z})$ in $H^1(X;mathbb {Z}/2mathbb {Z})$. [ABSTRACT FROM AUTHOR]

In this note we study the homology of configuration spaces of closed manifolds. We sharpen the eventual periodicity results of Nagpal and Cantero-Palmer, prove integral homological stability for configuration spaces of odd-dimensional manifolds and introduce a stabilization map on the homology with Z[1/2]-coefficients of configuration spaces of odd-dimensional manifolds. [ABSTRACT FROM AUTHOR]

Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincaré dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-selfdual SO(3) connections on a closed four-manifold. This should be a useful tool in comparing gauge theoretic invariants of smooth four-manifolds. [ABSTRACT FROM AUTHOR]

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes the non-simply connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology. [ABSTRACT FROM AUTHOR]

The article discusses Heegaard Floer homology which is an oriented three-manifolds that use Heegaard diagrams of the three-manifold. It states that the construction of a suitable variant of Lagrangian Floer homology is in symmetry of a Heegaard surface. It notes that the null-homologous knot Floer homology detects the genus of the knot and are generalized to the case of links which give a multi-graded Abelian group.