Probing L-S category with maps

Abstract:

For any map $X\xrightarrow {f}Y$, we introduce two new homotopy invariants, ${\text {dcat}}\;f$ and ${\text {rcat}}\;f$. The classical category ${\text {cat}}\;f$ is a lower bound for both, while ${\text {dcat}}\;f \leq {\text {cat}}\;X$ and ${\text {rcat}}\;f \leq {\text {cat}}\;Y$. When $Y$ is an Eilenberg-Mac Lane space, $f$ represents a cohomology class and ${\text {dcat}}\;f$ often gives a good estimate for ${\text {cat}}\;X$. We prove that if $\Omega \in {H^n}(M;\mathbb {Z})$ is the fundamental class of a compact, simply connected $n$-manifold, then ${\text {dcat}}\;\Omega = {\text {cat}}\;M$. Similarly, when $X$ is sphere, then $f$ is a homotopy class and while ${\text {cat}}\;f = 1$, ${\text {rcat}}\;f$ can be a good approximation to ${\text {cat}}\;Y$. We show that if $\alpha \in {\pi _2}(\mathbb {C}{P^n})$ is nonzero, then ${\text {rcat}}\;\alpha = n$. Rational analogues are introduced and we prove that for $u \in {H^\ast }(X;\mathbb {Q})$, ${\text {dcat}_0}\;u = 1 \Leftrightarrow {u^2} = 0$ and $u$ is spherical.