HHomotopies and the universal fixed point property
Markus SzymikOctober 2013
A topological space has the fixed point property if every continuous self-map ofthat space has at least one fixed point. We demonstrate that there are seriousrestraints imposed by the requirement that there be a choice of fixed points thatis continuous whenever the self-map varies continuously. To even specify theproblem, we introduce the universal fixed point property. Our results apply inparticular to the analysis of convex subspaces of Banach spaces, to the topologyof finite-dimensional manifolds and CW complexes, and to the combinatorics ofKolmogorov spaces associated with finite posets.2010 MSC: 54H25, 55M20, 47H10.
Every continuous self-map of a non-empty, compact, convex subspace of a topologicalvector space has at least one fixed point. This theorem of Brouwer-Schauder-Tychonoffis a milestone of fixed point theory, and the applications of it are numerous. Wheneverthe existence of fixed points is guaranteed for every self-map, the space is called a fixedpoint space. This property is shared by many other spaces besides the ones alreadycovered by that famous result.But, what if the self-map varies continuously? Is there also a continuous choice offixed points? This is the question that we address here. It is surprising that there is noprevious literature on this topic in the general topological setting, despite the propertybeing clearly desirable: in practice, it is genuinely less useful to know the existenceof a solution to a problem, if this solution does not depend continuously on the initialconditions. The Banach fixed point theorem provides a class of examples where we doknow that the fixed points depend continuously on the initial conditions (Example 2.1).However, the result evidently applies to metric spaces and contractions only.1 a r X i v : . [ m a t h . GN ] O c t dditional requirements on fixed point spaces that guarantee the existence of continu-ous choices of fixed points in general are made precise here and lead up to the notion of universal fixed point spaces (Definition 5.1). Among other things, we give a characteri-zation of these in terms of mapping spaces (Theorem 7.3) which is also the basis for thediscussion of some of our later examples. We show that the universal fixed point prop-erty is rather strong. The only convex subspaces of Banach spaces which are universalfixed point spaces are singletons (Theorem 4.5), and similarly for finite-dimensionalmanifolds and CW complexes (Corollary 4.8). Nonetheless, many examples of univer-sal fixed point spaces other than singletons do exist, and some of them are not evencontractible. For example, there are Kolmogorov spaces associated with finite posetsthat have these properties (Example 8.3).Unless otherwise stated, all vector spaces will be over the scalar field R of real numbers. In this section, we recall the basic definitions and some immediate consequences thatwe need.
Definition 1.1. (Fixed point property)
A topological space X has the fixed point prop-erty , if every continuous self-map f : X → X has a fixed point. One also says that X isa fixed point space . Example 1.2.
By Brouwer’s theorem [Bro12], all disks D n = { x ∈ R n |(cid:107) x (cid:107) (cid:54) } arefixed point spaces.The fixed point property has a long history, see [Bin69, p.120] for a brief and readableaccount of the origins of Brouwer’s result. Some years later, Schauder conjectured thatall non-empty, compact, and convex subspaces of topological vector spaces have thefixed point property, but he could only prove it for Banach spaces, see [Sch30]. Not longthereafter, Tychonoff, in [Tyc35], generalised this result to locally convex topologicalvector spaces. Only rather recently however, in a truly remarkable work [Cau01], RobertCauty proved Schauder’s conjecture in full generality, so that we are now able to record: Example 1.3.
All non-empty, compact, convex subspaces of topological vector spaceshave the fixed point property.Contrary to what the preceding examples might suggest, there are also many interestingtopological spaces which do not have the fixed point property.2 xample 1.4.
None of the spheres S n − = { x ∈ R n |(cid:107) x (cid:107) = } for n (cid:62) n =
0) is not a fixed point set. The spheresare in some sense the opposite of the contractibles. The first example of a compactcontractible space which does not have the fixed point property is due to Kinoshita,see [Kin53].Evidently, when X and Y are homeomorphic, and Y is a fixed point space, then so is X .Thus, being a fixed point space is a homeomorphism invariant. It is not a homotopyinvariant, as not all contractible spaces are fixed point spaces, while singleton spacesare.We reproduce a well-known proposition and its corollaries for the purpose of later gen-eralization. Proposition 1.5.
All topological spaces X that are retracts of a fixed point space Y , inthe sense that there are two continuous maps s : X → Y and r : Y → X such that rs = id X ,are fixed point space as well. Corollary 1.6.
Fixed point spaces are connected.
Corollary 1.7.
For any product X × Y that is a fixed point space, also both factors Xand Y are fixed point spaces.
The converse does not hold. The paper [Hus77] contains examples where X and Y aremanifolds. See also [Bro82] and the references therein for other examples.We close this section with a result on separation properties of fixed point spaces. Fixedpoint spaces need not be Hausdorff spaces, but they necessarily have to satisfy someweaker separation axiom, as the next proposition shows. Recall that a topological spaceis a Kolmogorov space if and only if for every pair of distinct points, at least one of themhas an open neighbourhood which does not contain the other.
Proposition 1.8.
All topological spaces that have the fixed point property are Kol-mogorov spaces.Proof.
By definition, if X is not a Kolmogorov space, then it contains two points x (cid:54) = x (cid:48) such that all open subsets of X contain either both of them or none of them. This meansthat the self-map f : X → X which sends x to x (cid:48) and everything else to x is continuous.And f does not have a fixed point. 3 Continuous families of fixed points
We are now addressing the question that forces itself on us: How do the fixed points ofa continuous self-map depend upon the given self-map? The following is a well-knownpositive example, where the fixed points also vary continuously.
Example 2.1.
Let ( X , d ) be a non-empty complete metric space, and let K be a realconstant with 0 (cid:54) K <
1. A self-map f : X → X that satisfies d ( f ( x ) , f ( y )) (cid:54) Kd ( x , y ) for all x and y is called a K-contraction on X . The Banach fixed point theorem [Ban22],in its usual form, says that every K -contraction f on X has a unique fixed point p = p ( f ) in X . Even more is true: The fixed points depend continuously on f ! More precisely,if ( f n ) is a sequence of K -contractions that converges uniformly to a K -contraction f ,then the fixed points p n = p ( f n ) converge to p = p ( f ) . In fact, the triangle equalityimplies d ( p n , p ) = d ( f n ( p n ) , f ( p )) (cid:54) d ( f n ( p n ) , f n ( p )) + d ( f n ( p ) , f ( p )) (cid:54) Kd ( p n , p ) + d ( f n ( p ) , f ( p )) , and this leads to the inequality d ( p n , p ) (cid:54) ( − K ) − d ( f n , f ) , so that the uniform conver-gence d ( f n , f ) → d ( p n , p ) → Definition 2.2. (Continuous family of self-maps)
Let X be a topological space. A continuous family of self-maps of X is a continuous map f : T × X −→ X , where T is another topological space. Remark 2.3.
Equivalently, a continuous family of self-maps of X is the second com-ponent of a continuous self-map ( id T , f ) of T × X which respects the projection to T ,the first one necessarily being the identity on T . This point of view has the advantageof being closer to the idea of a self-map, but the disadvantage of carrying the redundantcomponent id T around. However, this alternative suggests, as a generalization, to con-sider self-maps over T of total spaces E of fibre bundles E → T with fibre X over T .We do not pursue this idea here, as already the product situation turns out to be ratherrestrictive on X . 4 continuous family f : T × X → X of self-maps defines for each t in T a self-map f t : X −→ X , x (cid:55)−→ f ( t , x ) , on X which is continuous. We use this notation throughout this text. Definition 2.4. (Continuous family of fixed points)
Let f : T × X → X be a continuousfamily of self-maps of X . A continuous family of fixed points of f is a continuous map p : T −→ X such that(2.1) f ( t , p ( t )) = p ( t ) for all t in T .The condition (2.1) means that p ( t ) is a fixed point of f t for all t in T . Remark 2.5.
In the spirit of Remark 2.3, a continuous family p of fixed points of f isthe second component of a continuous section ( id T , p ) of the projection pr T to T suchthat ( id T , f )( id T , p ) = ( id T , p ) . Again, the first component necessarily is the identityon T . Example 2.6.
Just as every point of a topological space is a fixed point of somecontinuous self-map, for example of the constant self-map, or of the identity, so isevery continuous map p : T → X a continuous family of fixed points for some con-tinuous family f : T × X → X of self-maps. For example, take the family f definedby f ( t , x ) = p ( t ) , which is the family of constant self-maps, of course, or f ( t , x ) = x ,the constant family where every map f t is the identity id X .Example 4.4 shows that continuous families of fixed points need not exist in general. Definition 2.7. (Fixed point property with respect to a topological space)
A topo-logical space X has the fixed point property with respect to a topological space T iffor all continuous families f : T × X → X of self-maps of X , there exists at least onecontinuous family p : T → X of fixed points. Examples 2.8.
Trivially, if T is empty, then all topological spaces X have the fixedpoint property with respect to T . A topological spaces X has the fixed point propertywith respect to a singleton if and only if it is a fixed point space. More generally, thesame statement holds true when we replace the singleton by any non-empty discretespace. 5n the rest of this text, we consider the cases when X has the fixed point property withrespect to T = I , the unit interval, and the case when X has the fixed point property withrespect to all topological spaces, respectively. Before we do so, we prove some helpfulresults on retracts. In this short section, we prove that the fixed point property of a topological space X withrespect to a topological space T behaves well under retracts in T and in X . Proposition 3.1.
If a topological space X has the fixed point property with respect toa topological space T , and S is a retract of T , then X also has the fixed point propertywith respect to S.Proof.
Choose continuous maps j : S → T and r : T → S that satisfy the condi-tion r j = id S . If f : S × X → X is a continuous family of self-maps of X parametrizedby S then consider the continuous family g = f ( r × id X ) of self-maps of X parametrizedby T . By hypothesis of the proposition, it has a continuous family q : T → X of fixedpoints. It is now easy to check that p = q j a continuous family of fixed points for f .Proposition 1.5 and its corollaries transfer to the present context as follows. Proposition 3.2.
If a topological space X is a retract of a topological space Y whichhas the fixed point property with respect to a topological space T , then X also has thefixed point property with respect to T .Proof.
If the topological space X is a retract of the topological space Y , then wechoose continuous maps s : X → Y and r : Y → X that satisfy the condition rs = id X .Let f : T × X → X be a continuous family of self-maps of X . We now consider the con-tinuous family g = s f ( id T × r ) of self-maps of Y . By hypothesis, it has a continuousfamily q of fixed points. This means(3.1) s f ( t , rq ( t )) = q ( t ) for all t in T . It is then easy to check that p = rq is a continuous family of fixed pointsof f : Apply r to (3.1) and use the retraction property rs = id X .6 The homotopy fixed point property
In this section we discuss the behavior of fixed points with respect to homotopies.
Definition 4.1. (Homotopy fixed point property)
We say that a topological space X has the homotopy fixed point property if it has the fixed point property with respect tothe unit interval T = I = [ , ] .This means that a topological space X has the homotopy fixed point property if for allhomotopies f : I × X → X in X there is a continuous path p : I → X such that p ( t ) is afixed point of f t for all t in I . Remark 4.2.
We take care not to call a space which has the homotopy fixed pointproperty a homotopy fixed point space , as this terminology conflicts with a differentidea with the same name.There are many other examples of topological spaces which have the homotopy fixedpoint property, even non contractible ones. But, for the rest of this section we focus oncounterexamples.
Proposition 4.3.
All topological spaces that have the homotopy fixed point property arefixed point spaces.Proof.
This result is an immediate consequence of Proposition 3.1, since a singleton isa retract of the unit interval.The converse does not hold: the homotopy fixed point property is more restrictive thanthe ordinary fixed point property, as this example shows:
Example 4.4.
Continuous families of fixed points need not exist. Let X be the fixedpoint space [ , ] , and let T be the space [ , ] , both homeomorphic to the unit interval.Then f t ( x ) = x (cid:54) tx t (cid:54) x (cid:54) t + t + (cid:54) x is a continuous family of self-maps. 7igure 1: Graphs of the functions f t for t = / t =
1, and t = / f t are { x ∈ [ , ] | f t ( x ) = x } = { } t < [ , ] t = { } t > . This makes it clear that there is no continuous family of fixed points.The preceding example shows that the unit interval, which is a fixed point space, doesnot have the homotopy fixed point property. More generally, the n -disk D n does nothave the homotopy fixed point property as soon as n (cid:62)
1: If D n , for some n (cid:62)
1, had thehomotopy fixed point property, so had D , as D is a retract of D n . Still more general isthis theorem: Theorem 4.5.
A convex subspace X of a Banach space V has the homotopy fixed pointproperty if and only if it is a singleton.Proof.
The statement is clear for the singleton. So it is if X is empty. If X has atleast two points, let s : I → V be the straight path which connects the two points. TheHahn-Banach theorem provides for a map r : V → I such that the composition of therestriction of r to X with s is the identity, so that I is a retract of X . Since I does nothave the homotopy fixed point property by the preceding Example, nether does X byProposition 3.2. Example 4.6.
Recall that one model for the Hilbert cube is the set of sequences ( x n ) of real numbers such that | x n | (cid:54) / n for all n . This makes it clear that it is a convexsubset of the Hilbert space (cid:96) . It is homeomorphic to the countably infinite (Tychonoff)product of compact intervals, hence compact. By the preceding theorem, the Hilbertcube does not have the homotopy fixed point property.8t is unknown whether Theorem 4.5 generalizes to arbitrary topological vector spaces.With a little technique, we are able to implant the previously constructed counterexam-ples in most finite-dimensional topological spaces. Recall that for n (cid:62)
0, the n -ball isthe subspace B n = { x ∈ R n |(cid:107) x (cid:107) < } of euclidean space R n . Proposition 4.7.
All topological spaces X admitting an open embedding B n → X forsome n (cid:62) do not have the homotopy fixed point property.Proof. By Propositions 3.2 and Theorem 4.5, it suffices to show that D n is a retract ofthe topological space X . In order to prove this, we rescale the given open embedding soas to obtain another open embedding e : { x ∈ R n | (cid:107) x (cid:107) < } −→ X . Consider the continuous map λ : [ , ] → [ , ] with λ ( t ) = t t (cid:54) − t (cid:54) t (cid:54)
20 2 (cid:54) t . Then the continuous map f : { x ∈ R n | (cid:107) x (cid:107) < } −→ D n , x (cid:55)−→ λ ( (cid:107) x (cid:107) ) x extends by zero over e to a continuous map r : X → D n which restricts to the identityon D n . Corollary 4.8.
Finite-dimensional topological manifolds and finite-dimensional CWcomplexs do not have the homotopy fixed point property, unless they are singletons.Proof.
If the dimension of a finite-dimensional topological manifold or a finite-dimen-sional CW complex X is positive, then it admits an open embedding B n → X as requiredby the preceding proposition. If the dimension is zero, then X is discrete, and a discretespace which has the homotopy fixed point property must be a singleton. In this section we define the class of spaces where there is a continuous choice of fixedpoints whenever the self-map varies continuously.9 efinition 5.1. (Universal fixed point property)
We say that a topological space X has the universal fixed point property if it has the fixed point property with respect to alltopological spaces T .Clearly, the universal fixed point property implies the homotopy fixed point property,which implies the fixed point property. It is therefore a strong condition for a topologicalspace to be a universal fixed point space. For example, we already know that a convexsubspace X of a Banach space V does not have the universal fixed point property unlessit is a singleton (Theorem 4.5). If X is a finite-dimensional topological manifold, orif X is a finite-dimensional CW complex, then X is not a universal fixed point space,unless X is a singleton. (Corollary 4.8).In contrast to what these classes of non-examples suggest, singletons are not the onlyuniversal fixed point spaces, and there are even non-contractible examples. We identifythese examples using the concept of selection maps, to which we turn now. As we need some results on mapping spaces, let us briefly review these. See [Mun00]and [LS09], for example.Let X and Y be topological spaces. The subsets { f : X → Y | f ( K ) ⊆ V } , for K ⊆ X compact, and V ⊆ Y open, generate a topology on the set C ( X , Y ) of continu-ous maps X → Y , the compact-open-topology. If X is a compact Hausdorff space, and Y is metric, then this is the topology of uniform convergence. Some useful properties ofthe compact-open-topology in general are as follows: A continuous map f : T × X −→ Y defines for each t in T a continuous map f t : X −→ Y , x (cid:55)−→ f ( t , x ) , from X to Y . This yields a map f : T −→ C ( X , Y ) , t (cid:55)−→ f t , C ( X , Y ) , the so-called adjoint of f , which is continuous when we equip thetarget with the compact-open-topology. The map C ( T × X , Y ) −→ C ( T , C ( X , Y )) , f (cid:55)→ f , itself is clearly injective, but it is only surjective, for example, if X is a locally compactHausdorff space. In particular, the evaluation map C ( X , Y ) × X −→ Y , ( f , x ) (cid:55)−→ f ( x ) is continuous when X is a locally compact Hausdorff space.For every continuous family f : T × X → X of self-maps of X such that T is a locallycompact Hausdorff space, there is a continuous self-map(6.1) C ( T , X ) −→ C ( T , X ) , p (cid:55)−→ ( t (cid:55)→ f ( t , p ( t ))) . This proposition is useful since it reduces questions about continuous families of fixedpoints to question about (ordinary) fixed points:
Proposition 6.1.
If T is topological space which is a locally compact Hausdorff space,then a continuous family f : T × X → X of self-maps of X has a continuous family offixed points if and only if the self-map (6.1) of the mapping space C ( T , X ) has a fixedpoint.Proof. This result follows immediately from spelling out the definitions. If theself-map (6.1) has a fixed point p , then p : T → X is a continuous map suchthat f ( t , p ( t )) = p ( t ) , and this is Definition 2.4 of a continuous family of fixed pointsof f . And conversely, a continuous family of fixed points is a fixed point of that self-map. Corollary 6.2.
For all locally compact Hausdorff spaces T such that the mappingspace C ( T , X ) is a fixed point space, the space X has the fixed point property withrespect to T . Not all self-maps of C ( T , X ) need to have the form (6.1), so that we cannot infer herethat the converse of the corollary also holds. Example 6.3.
For the unit interval I = [ , ] , the space C ( I , I ) of self-maps is not afixed point space. This claim follows immediately from Corollary 6.2 and Example 4.4.But, of course, it is also easy to see this explicitly. For example, the continuous self-map f : C ( I , I ) → C ( I , I ) such that f ( v ) : t (cid:55)→ tv ( t ) does not have a fixed point.11t is useful to know that if X is a retract of Y , then C ( X , X ) is a retract of C ( Y , Y ) . Weomit the proof, as we do not need this result. In this section, we give a criterion for topological spaces to have the universal fixedpoint property. Here is the definition that lays the basis for it.
Definition 7.1.
Let X be a topological space. We call a continuous map Φ : C ( X , X ) −→ X with the property that Φ ( f ) is a fixed point of f , f ( Φ ( f )) = Φ ( f ) , for all f in C ( X , X ) , a selection map for X .A selection map for X continuously selects a fixed point Φ ( f ) for every self-map f of X . A similar concept has been introduced (much earlier, but independently) in arelated context in [DS87]. Proposition 7.2.
Let X be a topological space that admits a selection map. Then theproduct X × Y is a fixed point space for all fixed point spaces Y , and it admits a selectionmap for all Y that admit selection maps.
This result corresponds to and generalizes Theorem 1 in loc. cit. . Before we begin withthe proof, let us introduce some notation in connection with a continuous self-map f : X × Y −→ X × Y on a space that is a product of two other spaces X and Y . For given x in X and y in Y ,there are continuous maps f X , y : X −→ X , x (cid:55)−→ pr X f ( x , y ) , and similarly f x , Y : Y → Y . The pair ( x , y ) is a fixed point of f if and only if both x is afixed point of f X , y and y is a fixed point of f x , Y . The two maps depend continuously onthe given points, and this defines maps f X , ? : Y −→ C ( X , X ) , y (cid:55)−→ f X , y , and similarly f ? , Y : X → C ( Y , Y ) . 12 roof. We choose a selection map Φ : C ( X , X ) → X for X .Let f : X × Y → X × Y be a continuous self-map. We have to show that it has a fixedpoint. To do so, consider the self-map f Φ Y : Y −→ Y , y (cid:55)−→ f Φ ( f X , y ) , Y ( y ) . If Y is a fixed point space, then this self-map has a fixed point y . Then x = Φ ( f X , y ) isa fixed point of f X , y and y is a fixed point of f x , Y , so that ( x , y ) is a fixed point of f , asrequired.If Y also has a selection map Ψ , then we use it to define y = Ψ ( f Φ Y ) as a function of f ,and produce a selection map for X × Y from that, as in the first part of the proof.It follows from Proposition 7.2 that topological spaces with a selection map are fixedpoint spaces (take Y a singleton). But, more it true: As the main result of this section,we present a useful criterion for a topological space to have the universal fixed pointproperty. Theorem 7.3.
All topological spaces X that admits a selection map have the universalfixed point property. The converse holds for all topological spaces X for which theevaluation map C ( X , X ) × X → X is continuous.Proof.
Let f : T × X → X be a continuous family of self-maps of X parametrized bysome topological space, and let again f : T → C ( X , X ) denote its adjoint. Then it iseasy to check that the composition p = Φ f is a continuous family of fixed points of f .If X is a universal fixed point space, then there exists a continuous family p of fixedpoints for each continuous family f of self-maps of X . If the evaluation map is con-tinuous, then we take it as f , and see that it has a continuous family C ( X , X ) → X of fixed points, which we denote by Φ . The fixed point equation f ( t , p ( t )) = p ( t ) reads ev ( g , Φ ( g )) = Φ ( g ) in this case, which means g ( Φ ( g )) = Φ ( g ) for all g , asdesired. Corollary 7.4.
A locally compact Hausdorff space X has the universal fixed point prop-erty if and only if it has the fixed point property with respect to the space C ( X , X ) ofself-maps of itself. With these criteria at hand, we now proceed to present many examples of universal fixedpoint spaces. 13
Spaces from partially ordered sets
By Proposition 1.8, every fixed point space is a Kolmogorov space. In this section, wedemonstrate the difficulties one encounters in the study of universal fixed point spacesby looking at examples with only finitely many points.The category of finite Kolmogorov spaces and continuous maps is isomorphic to thecategory of finite partially ordered sets and monotone maps, and the isomorphism is theidentity on underlying sets. See [Bir67], and [Sto66, Proposition 7]. Briefly, if X is aKolmogorov space, and x is an element in X , define U ( x ) to be the intersection of theopen subsets of X which contain x . When we write x (cid:54) x (cid:48) for U ( x ) ⊆ U ( x (cid:48) ) , then thisdefines a partial order (cid:54) on X , and U ( x (cid:48) ) = { x ∈ X | x (cid:54) x (cid:48) } . Conversely, if (cid:54) is a partialorder on X , then we define the Kolmogorov topology on X in such a way that the opensubsets U are those subsets which satisfy the property: if x (cid:54) x (cid:48) and U contains x (cid:48) , thenit contains x as well.In the context of posets, selection maps have been introduced in [DS87]. We now recordthat this concept precisely captures the universal fixed point property for the correspond-ing class of spaces. Proposition 8.1.
A finite poset admits a selection map if and only if its associated Kol-mogorov space is a universal fixed point space.Proof.
This follows from Theorem 7.3. The compact-open-topology on the set C ( X , X ) of continuous self-maps is the Kolmogorov topology associated with the point-wise par-tial order on maps: f (cid:54) g if and only if f ( x ) (cid:54) g ( x ) for all x in X . See [Sto66, Proposi-tion 9]. With respect to this order, evaluation is order preserving, hence continuous.The existence of selection maps has been shown to be indeed stronger than the fixedpoint property in [PR92]. From our perspective, this comes as no surprise, since thereare examples of fixed point spaces that are not universal. We now present examples ofuniversal fixed point spaces that are not contractible.Rival, in [Riv76], has introduced a notion of dismantlability (by irreducibles) for finiteposets, and this has been extended to arbitrary posets in [BB79]. We do not recall thedefinitions here, but rather record a characterization, which follows from earlier resultsin [Sto66]. Proposition 8.2.
A finite poset is dismantlable if and only if its associated Kolmogorovspace is contractible. S = { α , ω } with open subsets Ø, { ω } , and S is a universal fixed point space,as is any Kolmogorov space associated with a finite poset that has a minimal or maxi-mal element. However, the converse is not true: Not all finite Kolmogorov spaces thatare universal fixed point spaces are also contractible. Here is an example. Example 8.3.
Rival, also in [Riv76], has given an example of a finite poset with thefixed point property that is not dismantlable, so that its associated Kolmogorov space isnot contractible. The list in [Rut89] contains some more examples. All these exampleseven admit selection maps by [PRS91].Figure 2: A poset with a selection map that is not dismantlableTherefore, the associated Kolmogorov spaces have the universal fixed point property byProposition 8.1
Acknowledgments
This research has been supported by the Danish National Research Foundation throughthe Centre for Symmetry and Deformation (DNRF92). I also thank Michal Kukiela fora helpful correspondence that brought the references in Order to my attention.
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Mathematisches InstitutHeinrich-Heine-Universit¨at40225 D¨usseldorfGermany
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Department of Mathematical SciencesUniversity of Copenhagen2100 Copenhagen ØDenmark