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<http://topology.auburn.edu/tp/>

## 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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<http://topology.auburn.edu/tp/>

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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

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(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 &lt; 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 &lt; 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 &lt; 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).

3. 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,

6. 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.

10. [8J B. V. S. Thomas, Free topological groups, Gen. Top. and Its Appl. 4 (1974), 51-72.
11. [9J o. Wyler, Top categories and categorical topology, Gen. Top. and Its App1. 1 (1971), 17-28.

University of Cape Town Cape Town, South Africa

Memphis State University Memphis, Tennessee 38112
