DIAGRAM COHOMOLOGY AND ISOVARIANT HOMOTOPY 11
(1.8). Postnikov decompositions, R-localization, and other common tech-
niques of homotopy theory can be developed for the category of admissible G-
invariant Cat (P)-diagrams of G-CW subcomplexes of a G-CW complex.
This is a special case of the conclusions at the bottom of [DF, p. 131] (in this
connection also see [May]).B
The following special case of (1.8) will be particularly important for our pur-
poses:
(1.9). An analog of M0ller's equivariant Barratt-Federer spectral sequence is
valid for the homotopy groups of equivariant Cat (P)-function spaces for Cat(P)-
maps from an admissible Cat(P) finite G-CW subcomplex diagram X to a
Cat(P)-simple G-CW complex diagram Y.
This is true because the treatment of [M0] is basically a formal consequence
of the existence of skeleta for X and Postnikov systems for Y in the appropriate
category.
There is also a straightforward extension of the Fary spectral sequence of
Theorem 1.5 to diagram categories:
(1.10). An analog of the equivariant Barratt-Federer/Fary spectral sequence
in Theorem 1.5 is valid for the homotopy groups of equivariant Cat (P)-function
spaces for Cat (P)-maps from an admissible Cat(P)-imite G-CW complex diagram
X to a Cat(F)-simple G-CW complex diagram Y.
For the sake of completeness we note that diagram cohomology (of ordinary
spaces) with twisted coefficients is defined in [DK, Section 3]; as usual, one can
define cohomology groups with compact supports by taking colimits of suitable
relative groups.
REMARKS ON THE PROOF OF
(1.10). The setting of [Sc2] shows that a
bifiltration defines a Fary spectral sequence if certain vanishing conditions hold
(see [Sc2, Assumptions A and A', p. 75]). This condition is established for
the mixed skeletal and orbit type bifiltration of a finite simplicial G-complex in
[Sc2, Prop. 4.2, p. 82]; the same argument goes through for diagrams of finite
simplicial G-subcomplexes of a finite simplicial G-complex.•
In [DF] and [DF/Z] the homotopy groups of diagram function spaces are
also analyzed in terms of unstable Ext-type spectral sequences. Although this is
beyond the scope of the present paper, a comparison of the results in this paper
with those of [DF/Z] seems worthwhile and possibly quite illuminating.
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