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).