Equivariant CW-complexes and the orbit category

Abstract

Abstract. We give a general framework for studying G-CW complexes via the orbit category. As an application we show that the symmetric group G = S 5 admits a finite G-CW complex X homotopy equivalent to a sphere, with cyclic isotropy subgroups.

Group actions on spheres with rank one isotropy

“…This theorem will be proved by applying the realization statement of Theorem 2.6. To construct a suitable finite free chain complex C over ZΓ G , we work one prime at a time to construct local models C (p) , and then apply the glueing method for chain complexes developed in [11,Theorem 6.7].…”

“…Since R = Z (q) , if we take a q-subgroup Q ≤ N G (H)/H with H = 1, and restrict C (q) (H) to Q, we obtain a finite dimensional projective RQ-complex (see [11,Lemma 3.6]). This means Q has periodic group cohomology and therefore it is a rank one subgroup.…”

“…Now suppose that K ∈ J p is such that (K) is a maximal conjugacy class in F k+1 . Consider the RΓ G -complex E K P where E K denotes the extension functor defined in [11,Sect. 2C].…”

“…The above lemma shows that the homology of I k+1 P is exactly the RΓ G -module J k+1 that we would like to add to the homology of C. To construct D we use an idea similar to the idea used in [11,Section 9B]. Observe that for every RΓ G -chain map f : N → C, there is a push-out diagram of chain complexes…”

“…In Section 4, we add homology to these local models so that these modified local complexes C (p) all have exactly the same dimension function. Results established in [11] are used to glue these algebraic complexes together over ZΓ G , and then to eliminate a finiteness obstruction. In Section 5 we combine these ingredients to give a complete proof for Theorem 5.1 and Theorem A.…”