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Sören lUman

is a G-map then / is a simplicial G-map or a linear G-map if and only if the induced map J:K/G-*L/G is simplicial or linear, respectively.

Let jK be an equivariant simplicial complex and f-.K-^Sa G-map where S is a G-space, and let {t/Jfgj be a cover of S by open G-subsets of S. Then there exists an equivanant subdivision X' of К such that for any vertex v of K' we have f{GSt{v,K'))cUi, for some set t/^. This follows by applying the corresponding ordinary non-equivariant result (Whitehead [14], Theorem 35) in the orbit spaces. In particular we have for each G-simplex Gs that /(Gs)ci7„ for some [/;. Here St(v, K') denotes the (closed) star of v in K\

IÎX is a subset of К we denote by NQC, K) the simplicial neighborhood ofZ in K, i.e., NQC, K) is a subcomplex of К and is the union of all simplexes of К that intersect X. By NQC, K) we denote the union of all open simplexes s of X for which s nX + 0. Then NQC, K) is an open neighborhood of X in K. Let Я be a subgroup of G, and assume that X is an Я-subset of the G-equivariant simplicial complex K. Then N{X,K) is an Я-equivariant subcomplex of X, in particular N{x,K) is a G^- equivariant subcomplex of X.

Let Xi be an equi variant subcomplex of X and let K\ be an equivariant subdivision of X^. Then the standard extension ofK\ to a subdivision K' of К (see Munkres [8], p. 76, but observe that his notations are somewhat different from ours) is an equivariant subdivision of X. We refer to this simply by the expression ; the standard extension ofK\ to X. This equivariant subdivision X' of X is such that it equals X outside N{K^,K), i.e., every simplex of X that lies in |X|-iV(Xi,X) occurs as a simplex in K'.

Let X : G-^0(n) be an orthogonal representation of G. By R"{x) we denote euclidean space JR" together with G-action through t. Let X be a simplicial G-complex. A linear G-map /: K-*R%x) is a G-map which is a linear map from the simpUcial complex X into euclidean space R". If / moreover is an embedding, i.e., /:X->/(X) is a homeomorphism, then / is called a rectilinear G-imbedding of X into R"{r), and the subspace /(X), together with the simplicial structure induced

from the homeomoфhism /: K^f{K), is called a rectilinear simplicial G-complex in Я"(т). If X moreover is an equivariant simplicial complex we call /(X) a rectilinear equivariant simplicial complex in R"{t).

A rectilinear cell G-complex К in К"(т) is a rectilinear cell complex X in R" (Munkres [8], Definition 7.6) such that X is a G-subset ofR%x) and if с is a cell in X then gc is also a cell in X for every geG. A G-subdivision X' of a rectilinear cell G-complex X is a subdivision X' of the rectilinear cell complex X such that X' becomes a rectilinear cell G-complex. It follows inunediately that if Xj and X2 are rectilinear cell G-complexes in jR"(t) such that |XJ = |X2l then X^ and X2 have a common G-subdivision. Moreover any rectilinear cell G-complex X in R"{x) has a G-subdivision K' such that K' is a rectilinear shnplicial G-complex in ä"(t). The proof of this fact is the same as in the ordinary case, see e.g. Lemma 7.8 in Munkres [8], in fact choosing the interior points of cells at which starring takes place to be the centroids the process is automatically equivariant. By further taking the second barycentric subdivision of K' we get a G-subdivision of the rectilinear cell G- complex X which is a rectilinear equivariant simplicial complex in R"{x). It also follows that if Xj and X2 are rectilinear equivariant simplicial complexes in R%x)