Title: The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with
Even-Dimensional Cells for $G = \mathbb{Z}/p$
Authors: Kevin K. Ferland and L. Gaunce Lewis, Jr.
AMS Classification numbers: Primary 55M35, 55N91, 57S17; Secondary 14M15 55P91
Addresses: Department of Mathematics, Bloomsburg University, Bloomsburg, PA 17815 and
Department of Mathematics, Syracuse University, Syracuse NY 13244-1150
email: kferland@bloomu.edu
lglewis@syr.edu
Abstract: 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 generalized $G$-cell
complexes is, however, not obvious, since \roG-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 \roG-graded
equivariant ordinary homology \HoeX{G}{X}{A} is free as a graded module over the
homology \HoPt 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 \HoeX{G}{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 \roG-graded equivariant ordinary homology.
In order to employ the Universal Coefficient Theorem, we must introduce the box
product of \roG-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 \roG-graded
equivariant ordinary homology is presented in a separate part at the end of
the paper.