Equivariant regular neighbourhoods

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INTRODUCTIONIn this paper L. Siebenmann's theory of open regular neighbourhoods [7,8,9] is generalized to the equivariant case. As suggested in [7; §4] and [8; §6] the basic definitions and axiomatic properties of such neighbourhoods set out in §2 of this paper follow exactly the outline of the non-eqivariant case. The main non-trivial result here about equivariant regular neighbourhoods perse is an existence theorem based upon the fundamental existence theorem of [9]. The idea is to give conditions which guarantee that a subspace of a G-space admits equivariant regular neighbourhoods if it admits ordinary regular neighbourhoods. See Theorem 3.4 for a precise statement.The nicest immediate applications of the general theory are to semi free finite group actions with isolated fixed points on manifolds. In this case the existence theorem referred to above shows that (under suitable dimension restrictions) each fixed point is contained in arbitrarily small invariant open disk neighbourhoods.In §§4 and 5 further applications are made. These could have been handled directly by adhoc arguments in each case. But it is enlightening and no more difficult in the end to develop the unifying general theory of equivariant regular neighbourhoods first.In §4 it is shown (with some dimension restrictions) that the space of all actions of a finite group on a compact manifold (with the compact-open topology) is locally contractible at each semifree action with finite fixed point set.

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