# The \(RO(G)\)-Graded Equivariant Ordinary Homology of \(G\)-Cell Complexes with Even-Dimensional Cells for \(G=\mathbb {Z}/p\)

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*Kevin K. Ferland; L. Gaunce Lewis, Jr.*

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.

#### Table of Contents

# Table of Contents

## The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G=\mathbb{Z}/p$

- Contents v6 free
- Introduction 110 free
- Part 1. The Homology of Z/p-Cell Complexes with Even-Dimensional Cells 716 free
- Chapter 1. Preliminaries 817
- Chapter 2. The main freeness theorem (Theorem 2.6) 2635
- Chapter 3. An outline of the proof of the main freeness result (Theorem 2.6) 3443
- Chapter 4. Proving the single-cell freeness results 4655
- Chapter 5. Computing H[sup(G)][sub(*)] (B U DV; A) in the key dimensions 5867
- Chapter 6. Dimension-shifting long exact sequences 6978
- 6.1. Preliminary observations about dimension-shifting sequences 6978
- 6.2. The reduction to complexity one dimension-shifting sequences 7382
- 6.3. Sequences with minimal complexity and spread 7685
- 6.4. The reduction to sequences of minimal spread 7988
- 6.5. The congruence condition on d[sub((v+∑w[sub(i)] ∑w'[sub(j)])]) 8493

- Chapter 7. Complex Grassmannian manifolds 8695

- Part 2. Observations about RO(G)-graded equivariant ordinary homology 95104