Applications of sheaf theory to the computation of invariants of LS-category type - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:24:48Z http://mathoverflow.net/feeds/question/40923 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40923/applications-of-sheaf-theory-to-the-computation-of-invariants-of-ls-category-type Applications of sheaf theory to the computation of invariants of LS-category type Mark Grant 2010-10-03T11:01:10Z 2011-01-07T17:17:31Z <p>I would like to know if sheaf theory can be applied to a particular class of questions in topology. </p> <p>The <em>Schwarz genus</em> (also known as sectional category) of a continuous map $p\colon\thinspace E\to B$ is the smallest integer $k$ such that $B$ can be covered by open subsets $U_1,\ldots ,U_k$ over each of which $p$ admits a <em>local section</em>, ie a continuous map $s_i\colon\thinspace U_i\to E$ such that $p\circ s_i$ equals the inclusion $U_i\hookrightarrow B$. If no such cover exists (eg if $p$ is not surjective) we set the genus to be $\infty$.</p> <p>Several important numerical invariants in topology are special cases of this genus. For instance, the Lusternik-Schnirelmann category $\mathrm{cat}(X)$ of a space $X$, defined to be the smallest $k$ such that $X$ can be covered by open subsets $U_1,\ldots ,U_k$ such that each inclusion $U_i\hookrightarrow X$ is null-homotopic, is easily seen to be the genus of the Serre fibration $PX\to X$ of based paths on $X$. More recently, Farber has defined the <em>topological complexity</em> $\mathrm{TC}(X)$ of a space $X$ to be the genus of the fibration $X^I\to X\times X$ which takes a free path in $X$ to its pair of initial and final points, and this is relevant to the motion planning problem in robotics. </p> <p>On the other hand, one of the most natural examples of a sheaf (at least for a topologist) is the sheaf of sections of a continuous map $p\colon\thinspace E\to B$. This is the sheaf $\Gamma(p)$ on $B$ whose sections over an open set $U\subseteq B$ are the set $\Gamma(p)(U)$ of local sections $s\colon\thinspace U\to E$ of $p$, as defined above. </p> <p>This leads me to the following (perhaps naive) question. </p> <blockquote> <p><strong>Definition.</strong> Let $\mathcal{F}$ be a sheaf of sets over $X$. Define the <em>Schwarz genus</em> of $\mathcal{F}$ to be the least $k$<br> such that $X$ has a cover by open subsets $U_1,\ldots, U_k$ such that each $\mathcal{F}(U_i)\neq\emptyset$. </p> <p>Do there exist techniques in sheaf theory to approximate (bound from above or below) the Schwarz genus of $\mathcal{F}$? Has this invariant been considered before? </p> </blockquote> <p>I suspect the answer is no, but I would like to hear this from a sheafy person (of which there seem to plenty on MO). Also I would be interested to hear of any extra conditions<br> you would impose to make the question more interesting or tractable. </p> <p><strong>Edit:</strong> As Ben noted below, we can apply the free abelian group functor to $\mathcal{F}$ to get a sheaf $\mathcal{G}$ of abelian groups over $X$. Define the Schwarz genus of $\mathcal{G}$ to be the least $k$ such that $X$ has a cover by open subsets $U_1,\ldots, U_k$ such that each $\mathcal{G}(U_i)\neq 0$. Can we obtain bounds on this genus using the Cech spectral sequence, or something similar?</p>