The Cohomology of G-spaces (G compact group)

Thursday, April 30, 2015 -
3:00pm to 4:00pm
Please note special time.  P. A. Smith proved in the late 1930'sTheorem: If $G = (Z/p)^r$ acts freely on the sphere $S^n$, then $r <2$.  This leads to:Rank Conjecture: If $G$ acts freely on a product of spheres $X = (S^{n_1})  \times  \ldots  \times  (S^{n_k})$ , as the identity on $H^{*}$,  then $r  <  k+1$.If a group $G$ acts freely on a space  $X$  then there is  a map $f\colon X/G \longrightarrow B_G$ (the classifying space of $G$), making $H^{*}(X/G)$ into an $H^{*}(B_G)$-module.  For $G$ anelementary abelian  $p$-group (as above), $H^2(B_G) =$ the vector space over $F_p$ of rank  $r$ , and let  $y_1,\ldots, y_r$ be a basis.  Define  (the linear) map  $\phi \colon H^2(B_G)  \longrightarrow L  =$  direct limit of $H^{2p^{q}}(X/G)$ using the $p$-th power map. Define the nil rank of $X$ to be the rank of the kernel$(\phi)$.  If  $F \colon Y \longrightarrow X$ is a $G$-map, then nil rank of  $Y$ is at least as large as nil rank of $X$.  $H^{*}(X/G)$ is $H^{*}(B_G)$ nilpotent means nil rank $X = r$.   If  $H^{*}(X)$ is finitely generated then $H^{*}(X/G)$ is also if and only if nil rank of  $X  =  r$.  This of course is the case for a finite dimensional $G$-space $X$.  But infinite dimensional spaces may be studied and one might try to climb the Postnikov tower of $X$ to prove theorems, where the spaces are rarely finite dimensional.  For example we show:Theorem:  If $G$ acts $H^{*}(B/G)$ nilpotently (as above) on $X$ homotopy equivalent to $Y \times (S^1)^q$  then a subgoup $K$ of  $G$ of rank $\geq r - q$ acts $H^{*}(B/K)$ nilpotently on $Y$.  Corollary:  The rank conjecture holds for $X$ homotopy equivalent to a product of circles and  $S^n$'s (single $n$).Theorem:  Let $p= 2$ and let $G$ act $H^{*}(B/G)$ nilpotently on  $X$ of the homotopy type of $(S^3)^s  \times  (S^n)^t$  Then  $r \leq s + t$. A similar more complicated theorem holds for $p > 2$.
William Browder
Princeton University
Event Location: 
Fine Hall 314