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Rudyaks conjecture states that cat$(M) geq$ cat$(N)$ given a degree one map $f:M to N$ between closed manifolds. We generalize this conjecture to sectional category, and follow the methodology of [5] to get the following result: Given a normal map of degree one $f:M to N$ between smooth closed manifolds, fibrations $p^M:E^M to M$ and $p^N:E^N to N$, and lift $overline{f}$ of $f$ with respect to $p^M$ and $p^N$, i.e., $fp^M = overline{f} p^N$; then if $f$ has no surgery obstructions and $N$ satisfies the inequality $5 leq dim N leq 2r$ secat$(p^N) - 3$ (where the fiber of $p^N$ is $(r-2)$-connected for some $r geq 1$), then secat$(p^M) geq$secat$(p^N)$. Finally, we apply this result to the case of higher topological complexity when $N$ is simply connected.
Using the surgery we prove the following: THEOREM. Let $f:M to N$ be a normal map of degree one between closed manifolds with $N$ being $(r-1)$-connected, $rge 1$. If $N$ satisfies the inequality $dim N leq 2r cat N - 3$, then for the Lusternik-Schnirelmann category $cat M geq cat N$ .
We prove the topological analogue of the period-index conjecture in each dimension away from a small set of primes.
We show that the analogue of the Peterson conjecture on the action of Steenrod squares does not hold in motivic cohomology.
Let an n-algebra mean an algebra over the chain complex of the little n-cubes operad. We give a proof of Kontsevichs conjecture, which states that for a suitable notion of Hochschild cohomology in the category of n-algebras, the Hochschild cohomology
To a direct sum of holomorphic line bundles, we can associate two fibrations, whose fibers are, respectively, the corresponding full flag manifold and the corresponding projective space. Iterating these procedures gives, respectively, a flag Bott tow