We provide a translation between Chekanov’s combinatorial theory for invariants of Legendrian knots in the standard contact R and a relative version of Eliashberg and Hofer’s contact homology. We use this translation to transport the idea of “coherent orientations” from the contact homology world to Chekanov’s combinatorial setting. As a result, we obtain a lifting of Chekanov’s differential gr...

We develop a formalism for relative Gromov-Witten invariants following Li [14, 15] that is analogous to the Symplectic Field Theory of Eliashberg, Givental, and Hofer [2]. This formalism allows us to express natural degeneration formulae in terms of generating functions and re-derive the formulae of Caporaso-Harris [1], Ran [19], and Vakil [21]. In addition, our framework gives a homology theor...

The aim of this article is to introduce and study certain topological invariants for oriented, rational homology three-spheres Y . These groups are relatively Z-graded Abelian groups associated to SpinC structures over Y . Given a Heegaard splitting of Y = U0∪Σ U1, these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real...

We define a notion of virtual fundamental class that applies to moduli spaces in gauge theory and in symplectic Gromov-Witten theory. For universal moduli spaces over a parameter space, the virtual fundamental class specifies an element of the Čech homology of the compactification of each fiber; it is defined if the compactification is “thin” in the sense that its boundary has homological codim...

We use a variation of of a classical construction of A. Hatcher to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension (at least twice the base dimension plus 3). Using a variation of the Dwyer-Weiss-Williams smoothing theory which we explain in a separate joint paper with Bruce Williams [11], we associate...

In this article, we show that magnitude homology and path are closely related, give some applications. We define differentials MH k ℓ ( G ) ⟶ − 1 $\operatorname{MH}^{\ell }_k(G) \longrightarrow \operatorname{MH}^{\ell -1}_{k-1}(G)$ between homologies of a digraph $G$ , which make them chain complexes. Then its $\mathcal {MH}^{\ell }_k(G)$ is non-trivial homotopy invariant in the context ‘homoto...

Gabrielov introduced the notion of relative closure of a Pfaffian couple as an alternative construction of the o-minimal structure generated by Khovanskii’s Pfaffian functions. In this paper, use the notion of format (or complexity) of a Pfaffian couple to derive explicit upper-bounds for the homology of its relative closure.