Type: Article
Publication Date: 1974-09-01
Citations: 29
DOI: https://doi.org/10.2140/pjm.1974.54.277
In this paper it is shown that given any group π and any subgroup G of the centre of π there exists a O-connected CPΓ-Complex X with π ± (X) ^ π under an isomorphism carrying the Gottlieb subgroup G^X) of n,(X) onto G.Introduction* Let X be a O-connected CTF-Complex and x o eX.In [2] Gottlieb defined a certain subgroup G(X, x Q ) of the fundamental group 7Γi(X, x 0 ) of X at x 0 and studied some of its properties.Earlier [4] Jiang Bo Ju defined a subgroup G f of π^X, f(x 0 )) corresponding to any map /: X-> X.The group G f when / = Id x turns out to be precisely G{X, x 0 ) studied by Gottlieb.These groups play a role in Nielsen-Wecken theory of fixed point classes and were investigated by R. F. Brown, W. J. Barnier, etc.In [3] Gottlieb defined the higher dimensional analogues G n (X, x Q ) c ττ Λ (X, x Q ) of G(X, x 0 ) and studied their properties.For any path σ joiningThus one can talk of the nth.Gottlieb group G n (X) of X without reference to a base point.In [2] it is shown that always G X {X) is a subgroup of the centre of 7t x {X) and that if X is a K(π, 1) CW-Complex G^X) is precisely the centre of π.Given any sequence of groups (π^j,^ with π k aberian for k ^ 2 it is known that there exists a O-connected CTF-Complex X with π k (X) = π k for all k.A natural question that suggests itself is the following:Given a sequence of groups {π k ) k ^ with π k abelian for k ^ 2 and subgroups G k of π k under what conditions does there exist a O-connected CPF-Complex with τc k (X) = π k under isomorphisms carrying G k (X) onto G k Ί Though we do not attempt to solve this general problem in this paper, we prove the following THEOREM.Given any group π and a subgroup G of the centre of π there exists a O-connected CW-Complex X with ^(X) = π under an isomorphism carrying G X {X) onto G.Finally I wish to thank Professor W. Browder for some very profitable discussions I had with him in connection with this problem.l Discrete group actions* This section deals with some results 277 278 K. VARADARAJAN