6 GIORA DULA AND REINHARD SCHULTZ
Y satisfying the Default Hypothesis. Then f extends equivariantly to X if a
sequence of obstructions
#m(f)
BR
H% (X, A;
G
7Tm-l(Y))
is trivialM
THEOREM
1.2. (Classification Theorem) Two continuous equivariant maps
/o, /i : X » Y (with X, Y as in 1.1) are equivariantly homotopic if a sequence
of difference obstructions 5m(/o,/i)
BRHQ(X,A;
G^m(Y)) is trivialM
THEOREM
1.3. (Barratt-Federer Spectral Sequence) Assume the same hy-
potheses as before, and also assume X is finite-dimensional. Modulo some mildly
exceptional behavior in dimensions 1 and 0 there is a spectral sequence such that
Efj =
BRHG*
(X; GiTjiY))
and Efj gives a series for 7Ti+j(FG(X,Y)), where FG(X,Y) is the space of G-
equivariant continuous maps from X toY with the compact open topology.M
The exceptional behavior in dimensions 1 and 0 is analogous to the phenomena
that arise in the ordinary Barratt-Federer spectral sequences [Ba, Fed] for these
dimensions.
Proofs of Theorems 1.1-1.3 can be found in M0ller's article [M0].
The usefulness of ordinary obstruction theory depends largely upon the com-
putability of ordinary singular cohomology. Since there are many effective means
for carrying out such computations, one can view obstruction theory as a means
for decomposing a homotopy-theoretic problem into tractable algebraic pieces.
Unfortunately, Bredon cohomology groups are considerably more difficult to
compute than their nonequivariant counterparts, and therefore calculations with
Theorems 1.1-1.3 generally require some additional input. For example, one
can combine Theorems 1.1-1.3 with the following spectral sequence going from
ordinary cohomology (possibly with twisted coefficients) to Bredon cohomology:
THEOREM
1.4. Let X be a finite G-CW complex whose isotropy subgroups lie
in a family ri satisfying the conditions for a list in [Wi], and let A be a system
of equivariant twisted coefficients over H. Choose an indexing {(Ki)} of the
conjugacy classes in H such that i j if a representative for (Kj) is contained
in some representative for (Ki). For each K e ri let SingFix(if,X) be the
G-subcomplex of Fix(K, X) consisting of all points whose isotropy subgroups
strictly contain K; denote the orbit spaces of these subcomplexes under the
action ofW(K) = N(K)/K by SingFix*(#,X) and Fix*(K,X) respectively
Then there is a spectral sequence such that
E\j
= #»+'" (Fix*(^-, X) - SingFix*^-, X); A(Kj)),
where H* refers to cohomology with compact supports and twisted coefficients,
the subgroup Kj represents (Kj), and E7^ gives a finite series for BRH%G3(X; A).
Remarks. (1) The twisting of coefficients in the
E1
term is determined by the
equivariant twisting data for the principal fibering of Fix(Kj, X)—SingFix(Kj, X)
over Fix*(i^,X) - SingFix*(i^,X) with fiber W(Kj).
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