I have this theoreme from the paper: J. Liu, The Morse index of a saddle point, 1989
My first question is what is $\tau$ is $\tau$ a chain ? so $I_m$ is the standard simplex ? if it is this why $\partial \tau$ has an image in $Y\cap \partial B_{\rho}$ ? generaly if $\tau: \Delta_p\rightarrow X$ then $\partial \tau: \Delta_{p-1}\rightarrow X$
Have you an idea please ?
Thank you.
I don't have access to the paper, but I think I can help:
I imagine that $I_m$ refers to an $m$-cell (or equivalently up to homotopy, an $m$-simplex). What's going on here is that $Y$ is filling the role of the descending manifold, for homological purposes, it seems.
Being finite-dimensional, $Y$ intersected with the ball is an $m$-ball, and $\tau$ is simply a standard map from the $m$-cell (itself an $m$-ball) to $Y$. Then the boundary of $\tau$ maps to the boundary of this $m$-ball, and generates the top homology of the boundary.
The remainder of the proof is devoted to showing that this is nonzero in the homology group $H_m(f_c \cap B_p, f_c \cap B_p - \{0\})$. Here I suppose $f_c = f^{-1}((-\infty,c])$, the sublevel set. The point is that we have $(I_m, I_{m-1}) \cong (B^m, S^{m-1}) \cong (Y\cap B_p,\partial (Y\cap B_p)) \subset (f_c \cap B_p, f_c \cap B_p - \{0\})$ and the rest of the proof shows this inclusion is nonzero on relative homology.