Having well-known isomorphism between groups K^0(X)\otimes R and H^{ev}(X;R), for compact space X, one can think of (generalized) homological theory related to K-theory same as homology theory of spaces for cohomology theory. This theory can be constructed in an abstract way by using topological N-dual space of X. M. F. Atiyah in his paper "Global Theory of Elliptic operators" showed how we can find representatives for elements of the theory, so called K-homology, by abstract elliptic operators on X. In this talk we will review atiyah's paper, it will include following parts:

- K-theory as a generalized cohomology theory and its appropriate way to define homology theory related to it.

- Elliptic operators on a compact manifolds and their abstract analogue for general compact spaces.

- K-index of elliptic operators and k-homology

We will show how to resolve the singularities of the symplectic orbifold $(CP^2(1,2,q), \omega_{FS})$, where $q > 0$ is an odd positive integer. The approach we take involves symplectic cutting, and this is the first step towards trying to understand the symplectomorphism group of the orbifold in question. A guiding philosophy is to try to keep the number of blow ups to a minimum (in this case, 3) since the symplectomorphism group becomes much harder to understand after a certain number of blow ups."