UNIVERSAL BASED-FREE INVOLUTIONS

By using a recent characterization result, we show that there are universal objects for various classes of based-free involutions.


Introduction
Let X and Y be spaces with involutions σ and τ , respectively. We say that a map f : (X, σ) → (Y, τ ) is equivariant provided that for every x ∈ X, τ (f (x)) = f (σ(x)).
Let σ Q , σ 2 and σ R ∞ be the standard involutions of Q, 2 and R ∞ , respectively, given by the formula x → −x.
An involution on a space is called based-free if it has a unique fixed-point. We prove that (Q, σ Q ) is universal for the class of all compact spaces with a based-free involution. That is, for every compact space X with based-free involution τ , there exists an equivariant embedding i : (X, τ ) → (Q, σ Q ). The compactness of X is essential, there does not exist an equivariant embedding ( 2 , σ 2 ) → (Q, σ Q ). We also show that ( 2 , σ 2 ) is universal for all spaces with based-free involutions. And, if the space under consideration is Polish, then the embedding can be chosen to be a closed embedding. The same results are not true for (R ∞ , σ R ∞ ): there does not exist an equivariant embedding ( 2 , σ 2 ) → (R ∞ , σ R ∞ ).

Extending involutions to convex sets
In this section, we show that any involution on a compact space X can be seen as the restriction of a 'convex' involution to X on some compact convex subset of R ∞ . We use a standard and well-known approach from functional analysis; it was used for example by Uspenskij [10] in his proof that the homeomorphism group of Q is universal for (separable metric) topological groups. Instead of using dual Banach spaces and weak * topologies, we give simple direct proofs of the facts that we need. We do so since describing the relevant results from functional analysis takes the same effort as proving these results directly. We stress that we use known methods that can be found in most textbooks in functional analysis, see e.g., [5, V, § §4,5].
With one exception, by space we mean separable metrizable topological space. An infinite-dimensional compact convex subset of a Fréchet space is called a Keller cube. It was shown by Keller [7], that every Keller cube is homeomorphic to Q. For every f : X → X we let Fix f denote its fixed-point set.
Let X be an infinite compact space, and let C(X) denote the Banach space of continuous realvalued functions on X.
Put K = {F ∈ R C(X) : F is continuous, linear, and F ≤ 1}. Then K is a convex subset of We claim that K is closed as well, and hence compact. (This is known of course, K is simply the unit ball in the dual space of C(X) endowed with the weak * topology). To prove this, fix G ∈ P \ K. We consider two cases. Assume first that G is not linear. Then there are f, g ∈ C(X) and λ, µ ∈ R such that is a neighborhood of G no element of which is linear. So we may assume without loss of generality that G is linear. Assume that there exists f ∈ C(X) such that , then G is continuous and has norm at most 1. This is a contradiction. Now let τ : X → X be an involution. We do not assume that τ is based-free, τ could be the identity. Defineτ : C(X) → C(X) byτ (f ) = f • τ . Thenτ is linear, which is trivial, and an isometry, which can be seen as follows. If f, g ∈ C(X) and p ∈ X, then for q = τ (p) we have Clearly,τ is an involution. Define e τ : R C(X) → R C(X) by e τ (F )(f ) = F (τ (f )). Then e τ simply permutes the coordinates of R C(X) , hence is a homeomorphism. It is also linear.
Letê τ : K → K be the restriction of e τ to K. Observe that since e τ is linear, e τ preserves convex combinations.
We next consider the natural embedding i : Proof. Indeed, for every f ∈ C(X), we havê as required.
Hence we can think ofê τ as an extension to K of the involution τ on X. We next claim that K is metrizable (this is also well-known of course), and hence is a Keller cube. (It is easy to see that K is infinite-dimensional, but we do not need this.) Observe that C(X) is separable, hence there is a countable set D which is dense in its unit ball. Consider the projection π : R C(X) → R D . Take arbitrary distinct F, G ∈ K. Then for some f ∈ C(X), F (f ) = G(f ). Since both F and G are linear, f = 0. Hence for g = f f we also have F (g) = G(g). Since g = 1, and since F and G are continuous, there consequently exists d ∈ D such that F (d) = G(d).
Enumerate D as {d n : n ∈ N} Since d n ≤ 1, for every n, this means that is an explicit admissible metric for K.
The same proof shows:

3.1.
A universal based-free involution for compact spaces. We will show that (Q, σ Q ) is universal for compact spaces with a based-free involution.
To this end, let X be a compact with a based-free involution τ , and let be its unique fixed-point. If X is finite, then the result is clear. So assume X is infinite. By the results in §2, we may assume that X is a subspace of a compact convex subspace K of R ∞ so that τ can be extended to an involutionτ on K which has the property that Fix(τ ) is convex and has arbitrarily smallτ -invariant open convex neighborhoods in K. We Put Y = K/ Fix(τ ). That is, we collapse the fixed-point set of K to a single point. Standard results imply that Y is an AR, [3, p. 121]. Moreover, the quotient map π : K → Y restricts to a homeomorphism on X. And Y has an induced involution σ that can be seen as an extension of the original involution τ on X. Since Fix(τ ) is convex and has arbitrarily small τ -invariant open convex neighborhoods in K, and for each such neighborhood U we have that U/ Fix(τ ) is an AR, we get that the unique fixed-point of σ has arbitrarily small open contractible invariant neighborhoods. Now put Z = Y × Q with involution τ Z = σ × σ Q . Then Z ≈ Q by Edwards' product theorem [4]. Clearly, the unique fixed-point of τ Z has arbitrarily small contractible τ Z -invariant neighborhoods. Hence (Z, τ Z ) is conjugate to (Q, σ Q ) by Wong's characterization theorem [11]. This completes the proof.
The compactness of X is essential since there does not exist an equivariant embedding e : ( 2 , σ 2 ) → (Q, σ Q ). For if there would exist such an embedding, the closure of e( 2 ) would be a compactification of 2 over which σ 2 can be extended to a based-free involution. But this is impossible by [9, §3].

3.2.
A universal based-free involution for all spaces. We will now show that ( 2 , σ 2 ) is universal for arbitrary spaces with a based-free involution.
Let X be a space with based-free involution σ. Let σ( ) = . We may again assume without loss of generality that X is infinite.
There is a compactification aX of X such that σ can be extended to an involution aσ : aX → aX.
By abuse of notation, we will denote aσ also by σ. Let F = Fix(σ). Then Let be any admissible metric on aX. Then d(x, y) = (x, y) + (σ(x),σ(y)) is admissible as well, andσ is an isometry with respect to d. This is the metric on aX that we will use in the sequel.
Defineσ : C(aX) → C(aX) byσ(f ) = f • σ. Thenσ is an involution and an isometry, see §2. Define i : aX → C(aX) in the standard way by The function i : X → C(aX) is an isometry, hence a topological embedding.
This is clear since i( ) is a fixed-point ofσ, andσ is an isometry.
Since C(aX) is an infinite-dimensional Banach space, known results imply that the U ε and V ε are all homeomorphic to 2 , [2, chapter VI]. Simply observe that U ε is convex and V ε ≈ C(aX) \ {0}.
Proof. Fix(σ) is a closed linear subspace of C(aX) of infinite codimension. Hence by the Bartle-Graves theorem, [2, p. 86], C(aX) is homeomorphic to the product of Fix(σ) and a Banach space of infinite dimension. In that Banach space the zero-vector is a Z-set (for example, since it is homeomorphic to R ∞ ). Hence the result follows.
Hence, indeed, ( 2 , σ 2 ) is universal. Observe that if X is Polish, then aX \ X is σ-compact. Since compact subsets in 2 are Z-sets, we can delete aX \ X as well. This gives us that for every Polish space X with a based-free involution σ, there is an equivariant closed embedding e : (X, τ ) → ( 2 , σ 2 ).