It is well known that the homology of a CW-complex with cells only in even dimensions is free. The equivariant analog of this result for $G$-cell complexes is, however, not obvious, since $RO(G)$-graded homology cannot be computed using cellular chains. This book considers $G = \mathbb{Z}/p$ and studies $G$-cell complexes.
It is well known that the cohomology of a finite CW-complex with cells only in even dimensions is free. The equivariant analog of this result for generalized G-cell complexes is, however, not obvious, since RO(G)-graded cohomology cannot be computed using cellular chains. We consider G = Z/p and study G-spaces that can be built as cell complexes using the unit disks of finite dimensional G-representations as cells. Our main result is that, if X is a G-complex containing only even dimensional representation cells and satisfying certain finite type assumptions, then the RO(G)-graded equivariant ordinary cohomology is free as a graded module over the cohomology of a point. This extends a result due to Gaunce Lewis about equivariant complex projective spaces with linear Z/p actions. Our new result applies more generally to equivariant complex Grassmannians with linear Z/p actions.
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It is well known that the homology of a CW-complex with cells only in even dimensions is free. The equivariant analog of this result for $G$-cell complexes is, however, not obvious, since $RO(G)$-graded homology cannot be computed using cellular chains. We consider $G = \mathbb{Z}/p$ and study $G$-cell complexes constructed using the unit disks of finite dimensional $G$-representations as cells. Our main result is that, if $X$ is a $G$-complex containing only even-dimensional representation cells and satisfying certain finiteness assumptions, then its $RO(G)$-graded equivariant ordinary homology $H_\ast^G(X;A>$ is free as a graded module over the homology $H_\ast$ of a point.This extends a result due to the second author about equivariant complex projective spaces with linear $\mathbb{Z}/p$-actions. Our new result applies more generally to equivariant complex Grassmannians with linear $\mathbb{Z}/p$-actions. Two aspects of our result are particularly striking. The first is that, even though the generators of $H^G_\ast(X;A)$ are in one-to-one correspondence with the cells of $X$, the dimension of each generator is not necessarily the same as the dimension of the corresponding cell. This shifting of dimensions seems to be a previously unobserved phenomenon. However, it arises so naturally and ubiquitously in our context that it seems likely that it will reappear elsewhere in equivariant homotopy theory. The second unexpected aspect of our result is that it is not a purely formal consequence of a trivial algebraic lemma.Instead, we must look at the homology of $X$ with several different choices of coefficients and apply the Universal Coefficient Theorem for $RO(G)$-graded equivariant ordinary homology. In order to employ the Universal Coefficient Theorem, we must introduce the box product of $RO(G)$-graded Mackey functors. We must also compute the $RO(G)$-graded equivariant ordinary homology of a point with an arbitrary Mackey functor as coefficients. This, and some other basic background material on $RO(G)$-graded equivariant ordinary homology, is presented in a separate part at the end of the memoir.
The key tools used are equivariant "even-dimensional freeness" and "multiplicative comparison" theorems for G-cell complexes, both proven by Lewis in [Lew88] and subsequently refined by Shulman in [Shu10], and with the former theorem extended by Basu and Ghosh in [BG16]. The latter theorem enables us to compute the multiplicative structure of the cohomology of BC2SU(2) by embedding it in a direct sum of cohomology rings whose structure is more easily understood. Both theorems require the cells of the G-cell complex to be attached in a well-behaved order, and a significant step in our work is to give BCnSU(2) a satisfactory Cn-cell complex structure.
