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Research Announcement
ON FREE TOPOLOGICAL GROUPS AND FREE PRODUCTS OF TOPOLOGICAL GROUPS
by
Temple H. Fay and Barbara Smith Thomas
Topology Proceedings
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Department of Mathematics & Statistics
Auburn University, Alabama 36849, USA
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ON FREE TOPOLOGICAL GROUPS AND FREE
PRODUCTS OF TOPOLOGICAL GROUPS
(summary of results to appear elsewhere)
Temple H. Fay and Barbara Smith Thomas
In attempting to characterize the epimorphisms in the category of Hausdorff topological groups, one is led to investi gating certain quotients of the free product G II G of a Haus dorff topological group with itself. In particular, one wants to know if the amalgamated free product G II G, for a B closed subgroup B, is Hausdorff, or if its Hausdorff reflection is "sufficiently large." To do this it seems useful to obtain information about the topological structure of G II G, or more generally of G II H for Hausdorff topological groups G and H.
Given any two topological groups (not necessarily Hausdorff) G and H, their coproduct G II H in the category of topological groups is G * H (their free product in the category of groups) with the finest topology compatible with the group structure making the injections G + G II H + H continuous (Wyler). The subgroup K of G II H generated by the reciprocal cowIDutators -1 -1 [g,h] ghg h is a normal subgroup and is freely generated. It is not difficult to show that every element of G II H has a unique representation ghc with c E K, and if G and Hare Hausdorff then K is closed in G II H. Theorem 4 below shows that if G and H are Hausdorff and if K can be given a suitable topology, then G II H is Hausdorff.
Since K is freely generated we begin with a short investi gation of Graev free groups.
Defn: The Graev free topological group over a pointed space (X,p) consists of a topological group FG(X,p) and a base point preserving continuous function n: (X,p) + FG(X,p) with the
usual unique factorization property
(the base point of a group is the identity, f is continuous and preserves base points, the induced f is a cortinuous group homo morphism, and the diagram commutes) .
Theorem (Wyler): The underlying group of FG(X,p) is the free group on X{p}~ and FG(X,p) has the finest topology compati ble with the group structure such that n:X + FG(X,p) is continu ous.
1 Theorem (Ordman): If X is a kw-space then FG(X,p) has the weak topology generated by the subsets [FG(X,p)]n = words of reduced length ~ n.
Theorem 1: FG(X,p) is Hausdorff if and only if X is func tionally Hausdorff.
Theorem 2: FG(X,p) contains X as a closed subspace if and only if X is Tychonoff.
Theorem 3: If X is Tychonoff and y is a closed subspace of X containing the base point then the subgroup of FG(X,p) gen erated by Y is closed.
We now turn to considering the topological structure of G II H for Hausdorff groups G and H.
Theorem 4: In order thdt G II H have a Hausdorff group
IX is a kw-space if it is the weak sum of countably many com pact Hausdorff spaces.
the topology it is necessary and sufficient that K ~ G IT H have a Hausdorff group topology~ no finer than~ but comparable to~ -2 topology of FG(G A H,e) such that W: (G x H) x K ~ K~ where -1 -1 W(g,h,c) ghch g ~ is continuous.
Note: the need for the continuity of W shows up a number of times in the proof of this theorem, of which the simplest example is (ghc)-l g-lh-l(hgh-lg- l ) (ghc-lh-lg- l ) g-lh-l[g,h]-lw(g,h,c- l )
Ordman's Theorem above says roughly "If X is a kw-space and something is true for words of length < n, for all n, then it is true for FG(X,p). Thus
Corollary (Katz): If G and H are kw-spaces~ then G IT H
is Hausdorff.
The essential observation in the proof is that for all n w : (G x H) x [FG(G A H,e] ~ F (G A H,e) is continuous.
The first answers a n n G
We conclude with a few observations. question of Morris, Ordman, and Thompson in the negative.
Observation 1: G A Hand [G,H] ~ G li H need not have the same topology: In X (wI+1) x wI the two closed sets A = {(x,x) lwo~x < WI} and B = ((wI+1) {I} ) U ({wI} wI) x x cannot be separated by open sets. Let G = FG(wl+l,w l ) and H = FG(wl,l). Then X is a closed subset of G x H, A is closep in G x Hand B = X n ((G x {e H }) U ({e G } x H)). Hence A and (G x {e }) U ({e } x H) cannot be separated by open sets. It H G follows that G A H is not even regular, and thus cannot be a subspace of G IT H.
Observation 2:
The method of proof used in the corollary
2G A H is the quotient of G x H obtained by collapsing (G x {eH}) U ({eG} x H) to a point, this collapsed point is denoted bye.
above cannot be used in the general case since FG(Q,O) does not have the weak topology generated by the words of length < n, and Q = Q 1\ {0,1}.
Obersvation 3: We cannot even replace FG(G 1\ H,e) by FG(S(G 1\ H) ,e) to obtain a proof of the general case since (Q x Q) x FG(S(Q 1\ Q) ,0) does not have the weak topology gener ated by the subsets (Q x Q) x [FG(S(Q 1\ n) ,O]n.
Partial Bibliography
- [1] M. I. Graev, Free topological groups, Izv. Akad. Nauk. SSSR Sere Mat. 12 ·(1948), 279-324, (Russian). Eng. Trans1.: Arner. Math. Soc. Trans1. 35 (1951), Reprint Arner. Math. Soc. Trans1. (1) 8 (1962), 305-364.
- [2] , On free products of topological groups, Izv. Akad. Nauk. SSSR Sere Mat. 14 (1950), 343-354, (Russian).
- Free Topological groups and principal fiber
- [3] E. Katz, bundles, Duke Journal 42 (1975), 83-90.
- [4] , Free products in the category of kw-groups,
- Pacific J. 59 (1975) 493-495.
- [5] S. Morris, E. T. Ordman, and H. B. Thompson, The topology of free products of topological groups, Proc. Second Internat. Conf. on Group Theory, Canberra, 1973, 504-515.
- [6] E. Ordman, Free products of topological groups which are kw-spaces, Trans. Amer. Math. Soc. 191 (1974), 61-73.
- [7] , Free k-groups and free topological groups, Gen. Top. and Its Appl. 5 (1975), 205-219.
- [8J B. V. S. Thomas, Free topological groups, Gen. Top. and Its Appl. 4 (1974), 51-72.
- [9J o. Wyler, Top categories and categorical topology, Gen. Top. and Its App1. 1 (1971), 17-28.
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