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.
We define and develop a homotopy invariant notion for the topological complexity of a map $f:X to Y$, denoted TC($f$), that interacts with TC($X$) and TC($Y$) in the same way cat($f$) interacts with cat($X$) and cat($Y$). Furthermore, TC($f$) and cat
($f$) satisfy the same inequalities as TC($X$) and cat($X$). We compare it to other invariants defined in the papers [15,16,17,18,20]. We apply TC($f$) to studying group homomorphisms $f:Hto G$.
Gravitational wave science is on the verge of direct observation of the waves predicted by Einsteins General Theory of Relativity and opening the exciting new field of gravitational wave astronomy. In the coming decades, ultra-sensitive arrays of gro
und-based instruments and complementary spaced-based instruments will observe the gravitational wave sky, inevitably discovering entirely unexpected phenomena while providing new insight into many of the most profound astrophysical phenomena known. in July 2007 the Gravitational Wave International Committee (GWIC) initiated the development of a strategic roadmap for the field of gravitational wave science with a 30-year horizon. The goal of this roadmap is to serve the international gravitational wave community and its stakeholders as a tool for the development of capabilities and facilities needed to address the exciting scientific opportunities on the intermediate and long-term horizons.
