Cohomology of manifolds

Question

Let $M$ be a manifold with the differential $d$ acting on exterior forms. A differential operator $\delta$ is defined by:

$$\delta (\alpha)= d \alpha \wedge \beta$$

with $\beta \wedge d \beta=0$, then we have $\delta \circ \delta =0$. The cohomology is defined:

$$H^*_{\beta}(M,{\bf R})=Ker(\delta)/Im(\delta)$$

Can we find new topological invariants of manifolds?