Conley index is a tool which allows to detect flow invariant subsets and in particular - in the case of gradient flows - critical points of the function. Starting with Wazewski's principle discovered in the 50's we motivate the definition of the Conley index. We then discuss why the usual definition of the cup-length is not suitable for applications and introduce the relative one. Finally, we announce a recent application of the relative cup-length to the degenerate Arnold conjecture on $T^{2n} \times CP^m$.