Suppose that $M$ has a proper Morse function $f\ge 0$ with all critical points of index at most $k$. Then homotopically $M$ can be made of low-dimensional cells: it has homotopical dimension at most $k$. But also $M$, relative to its boundary $f^{-1}(c)$ it M^{\ge c}$, can be made by attaching high-dimensional cells: . Thus the pair$(M,M^{\ge c})$is$(dim(M)-k-1)$-connected. So for example if$k\le dim(M)-3$and$M$is simply connected then$M$must also be "simply connected at infinity". 1 Suppose that$M$has a proper Morse function$f\ge 0$with all critical points of index at most$k$. Then homotopically$M$can be made of low-dimensional cells: it has homotopical dimension at most$k$. But also relative to its boundary$f^{-1}(c)$it can be made by attaching high-dimensional cells: the pair$(M,M^{\ge c})$is$(dim(M)-k-1)$-connected. So for example if$k\le dim(M)-3$and$M$is simply connected then$M\$ must also be "simply connected at infinity".