Seminar

1) M can be globally described as the zero locus of a differentiable section s of an infinite rank vector bundle E over an infinite dimensional manifold B. For instance M are stable pseudo-holomorphic maps from a given curve to a fixed almost complex manifold, B is the space of all differentiable maps. We assume ds(p) to be Fredholm of constant index d at every point p in M.

2) Locally on X, we can reduce B and E to finite dimension, i.e., replace them with a section s' of a finite rank bundle E' over a finite dimensional manifold B', in such a way that at every point, the differentials ds'(p) and ds(p) are quasi-isomorphic (in a sense to be made precise).

3) Finally we assume that the local finite dimensional reductions are unique up to "thickening" (again, in a sense to be made precise).

 

We discuss how, if ideas from algebraic geometry apply, points 1)-3) could be used to construct a cone stack in a vector bundle stack on M, which could lead to defining a virtual fundamental class in the homology of M.