Fibrewise infinite symmetric products and $M$-category

Bull. Korean Math. Soc. 1999 Vol. 36, No. 4, 671-682

Hans Scheerer and Manfred Stelzer Freie Universitat Berlin, Freie Universitat Berlin

Abstract : Using a base--point free version of the infinite symmetric product we define a fibrewise infinite symmetric product for any fibration $E\to B$. The construction works for any commutative ring $R$ with unit and is denoted by $R_{f}(E)\to B$. For any pointed space $B$ let $G_{i}(B)\to B$ be the $i$--th Ganea fibration. Defining $M_{R}$--$\!\cat(B):=\inf\{i\!\mid\! R_{f}(G_{i}(B))\to B$ admits a section\nolinebreak $\}$ we obtain an approximation to the Lusternik--Schnirelmann category of $B$ which satisfies e.g.\ a product formula. In particular, if $B$ is a $1$--connected rational space of finite rational type, then $M_{\Q}$--$\!\cat(B)$ coincides with the well--known (purely algebraically defined) $M$--category of $B$ which in fact is equal to $\cat(B)$ by a result of K.\ Hess. All the constructions more generally apply to the Ganea category of maps.