In this paper we obtain an explicit formula for the number of curves in P 2 , of degree d, passing through (d(d + 3)/2 − k) generic points and having a codimension k singularity, where k is at most 7. In the past, many of these numbers were computed using techniques from algebraic geometry. In this paper we use purely topological methods to count curves. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M , counted with a sign, is the Euler class of V evaluated on the fundamental class of M .

Abstract. We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold K ⊂ M . When the submanifold is a knot in R 3 , we show this structure recovers a specialization of Ng cord algebra [Ng3], a non-trivial knot invariant which is not determined by a number of other knot invariants.

In this paper we obtain an explicit formula for the number of curves in P 2 , of degree d, passing through (d(d + 3)/2 − (k + 1)) generic points and having one node and one codimension k singularity, where k is at most 6. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M , counted with a sign, is the Euler class of V evaluated on the fundamental class of M .

Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold are determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth four manifold with second Betti number at least three the number of geometrically distinct periodic geodesics of length at most l grows exponentially as a function of l. The number of closed Reeb orbits of length at most l on the spherization of T * M also grow exponentially for any Reeb flow.

No abstract