Rudyak's conjecture for lower dimensional 1-connected manifolds

Abstract: Rudyak's conjecture states that for any degree one map $f:M\to N$ between oriented closed manifolds there is the inequality $\cat (M)\ge \cat(N)$ for the Lusternik-Shnirelmann category. We prove the Rudyak's conjecture for $n$-dimensional simply connected manifolds for $n\le 8$.