Equivariant path fields on topological manifolds
aa r X i v : . [ m a t h . A T ] J un EQUIVARIANT PATH FIELDS ON TOPOLOGICALMANIFOLDS
LUC´ILIA BORSARI, FERNANDA CARDONA, AND PETER WONG
Abstract.
A classical theorem of H. Hopf asserts that a closed con-nected smooth manifold admits a nowhere vanishing vector field if andonly if its Euler characteristic is zero. R. Brown generalized Hopf’s re-sult to topological manifolds, replacing vector fields with path fields. Inthis note, we give an equivariant analog of Brown’s theorem for locallysmooth G -manifolds where G is a finite group. Introduction
Let M be a closed connected orientable smooth manifold. A classicaltheorem of H. Hopf [13] states that M admits a non-singular vector field ifand only if the Euler characteristic, χ ( M ), of M is zero. R. Brown [7] gavea generalization of Hopf’s theorem for topological manifolds, by replacingvector fields with path fields, a concept first introduced by J. Nash [22]. In[7], R. Brown showed that a compact topological manifold M admits a non-singular path field if and only if χ ( M ) = 0. Subsequently, R. Brown andE. Fadell [8] extended [7] to topological manifolds with boundary. It wasshown by E. Fadell [10] that any Wecken complex of zero Euler characteristicadmits a non-singular simple path field. R. Stern [24] showed the same resultfor topological manifolds of dimension different from four.The existence of a path field allows one to show the so-called CompleteInvariance Property (CIP) (see [17] and [23]). Recall that a topological space M is said to have the CIP if for any non-empty closed subset A ⊂ M , thereexists a map f : M → M such that A = F ix f := { x ∈ M | f ( x ) = x } . Date : September 18, 2018.2000
Mathematics Subject Classification.
Primary: 55M20; Secondary: 57S99.
Key words and phrases.
Equivariant Euler characteristic, equivariant path fields, lo-cally smooth G -manifolds.The third author acknowledges supported by a grant from the National ScienceFoundation. Similarly, M possesses the CIP with respect to deformation (denoted byCIPD) if f is homotopic to the identity 1 M . The non-singular path fieldproblem is equivalent to the fixed point free deformation problem. That is, M admits a non-singular path field if and only if 1 M is homotopic to a fixedpoint free map.In [18], [19], and [25], equivariant vector fields on compact smooth G -manifolds were studied. In particular, an equivariant analog of Hopf’s the-orem was proved in [18]. Furthermore, an equivariant analog of what wasdone for path fields on Wecken complexes in [10], was given in [26] and nec-essary and sufficient conditions for equivariant CIPD were given for smooth G -manifolds (see also [3] for a certain type of equivariant CIP). Similar tothe non-equivariant case, the equivariant non-singular path field problemis closely related to finding an equivariant fixed point free deformation. Itturns out that the existence of such a fixed point free map requires morethan merely the existence of non-equivariant fixed point free deformation onthe fixed point sets M H for each isotropy type ( H ) (see [11]).The objective of this paper is to prove an equivariant analog of Brown’stheorem [7] for topological manifolds with locally smooth action of a finitegroup G . Moreover, we extend the necessary and sufficient conditions for G -CIPD found in [27] to this category of G -manifolds.We would like to thank D.L. Gon¸calves and G. Peschke for very helpfulconversations and suggestions.Throughout G will always be a finite group acting on a compact space M where the action is locally smooth. For the definition and basic propertiesof locally smooth actions, we refer the reader to [5].2. Equivariant Euler characteristic and G -path fields In this section, we establish the necessary definitions of path fields andEuler characteristic in the equivariant category.Equivariant path fields were defined and studied in [26] and [27]. For ourpurposes, we think of G -path fields as sections of certain G -fibrations.First, given a G -map p : E → B , we say that p has the G - CoveringHomotopy Property ( G -CHP) if for all G -space X the following commutativediagram has a solution F : X × [0 , → E where all maps are G -equivariant. QUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS 3 X × { } f −−−−→ E incl. y y p X × [0 , H −−−−→ B A G -fibration is simply a G -map p : E → B satisfying the G -CHP for all G -spaces.Given a G - fibration p : E → B , we consider Ω p = { ( e, w ) ∈ E × B I | p ( e ) = w (0) } . Then Ω p is a G -invariant subspace of E × B I . Let e p : E I → Ω p bethe G -map defined by e p ( τ ) = ( τ (0) , p ( τ )). Consider the equivariant maps F : Ω p × I → B defined by F ( e, w, t ) = w ( t ), and f : Ω p → E by f ( e, w ) = e .Since p is a G -fibration, F can be lifted to a G -map e F : Ω p × I → B whichextends f . Then λ : Ω p → E I , defined by λ ( e, w )( t ) = e F ( e, w, t ), is anequivariant lifting function for p , that is, e p ◦ λ is the identity on Ω p .A G -fibration is called regular if it admits a regular G -lifting function,meaning, a G -lifting function satisfying λ ( e, p ( e )) = e , for all e ∈ E , where p ( e ) denotes the constant path at p ( e ). In [14], W. Hurewicz shows thatevery fibration over a metric space is regular. The same proof can be adaptedto the equivariant case, provided the metric d is assumed to be G -invariant,that is, d ( gx, gy ) = d ( x, y ), for all g ∈ G and x, y ∈ B . Lemma 2.1.
Let p : E → B be a regular G -fibration over a G -manifold B .Let ( X, A ) be a G -ANR pair and suppose that there are equivariant maps f : X × ∪ A × I → E and h : X × I → M such that p ◦ f = h | ( X × ∪ A × I ) .Then, there exists a G -map e f : X × I → E which extends f and such that p ◦ e f = h .Proof. This lemma is an equivariant version of Theorem 2.4 of [1]. The proofof this theorem in the non equivariant context is very constructive and itis possible to verify that, in all steps, we do obtain equivariant maps, aslong as we start with the appropriate equivariant setting and make use ofCorollary 2.3 of [25]. (cid:3)
Given a compact topological manifold M , the Nash path space T M of M consists of T M = { all constant paths } and the set T M of all paths α on M such that for 0 ≤ t ≤ α ( t ) = α (0) iff t = 0. Consider the map q : T M → M given by q ( α ) = α (0). With the compact-open topology on LUC´ILIA BORSARI, FERNANDA CARDONA, AND PETER WONG T M , the triple ( T M , q M , M ) is a Hurewicz fibration and the sections of q are called path fields on M . A path field is non-singular if it is a section in( T M , q M | T M , M ). A path field σ is simple if for any x ∈ M , σ ( x ) is a simplepath.If G acts on M , then G acts on T M via g ∗ α ( t ) = gα ( t ). Since q : T M → M is a fibration, it is straightforward to see that it is indeed a G -fibration wherethe G -action on [0 ,
1] is trivial. Thus, we define a G -path field to be a G -section s : M → T M of q so that q ◦ s = 1 M . Moreover, the subfibration q M : T M → M is also a G -subfibration. The notions of non-singular and ofsimple G -path fields are defined in the obvious fashion.Given a compact topological manifold M , the classical Euler characteristicof M is an integer and it coincides with the fixed point index of the identitymap 1 M . When a finite group G acts on M , the appropriate equivariantEuler characteristic takes the components of the various fixed point sets M H , H ≤ G , into account.We write | χ | ( M H ) = P C | χ ( C ) | , where C ranges over the connected com-ponents of M H = { x ∈ M | G x = H } . Here, G x denotes the isotropy sub-group of x . Since M is compact, each M H = { x ∈ M | hx = x, ∀ h ∈ M } isalso compact so that M H has only a finite number of components.3. Singularities of G -path fields In this section, we prove our main results following the approach of [7].Since we work in the G -manifolds category, many of the techniques employedin [7] must be modified for the equivariant setting, first of which is thefollowing relative equivariant domination theorem for compact G -ANRs. Theorem 3.1 (Relative Equivariant Domination Theorem) . Let M be an n -dimensional G − manifold and A be an invariant compact submanifold ofdimension k . We can find a G − complex K of dimension n , an invari-ant subcomplex of dimension k and equivariant maps ϕ : K −→ M and ψ : M −→ K , so that ψ is barycentric, ϕ | L : L −→ A , ψ | A : A −→ L , ϕ ◦ ψ ∼ = G id M and ϕ | L ◦ ψ | A ∼ = G id A Proof.
According to [2, Theorem 1], we can equivariantly embed M as aclosed G -neighborhood retract of a convex G -set in a Banach G -space A ( M )in which G acts isometrically. Now we follow the proof of the G -domination QUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS 5 theorem (Proposition 2.3) of [20]. Let r : O → M be the G -retractionof some G -invariant neighborhood O . It suffices to show that O can be G -dominated by a finite G -complex K . Let { W α } be a covering of O byconvex subsets which are open in O . Since M is compact, we can find afinite open covering {O α } of O such that the convex hull of a finite unionof the O β is contained in M . Then there is a finite open pointed G -covering V = { V γ , v γ } , a refinement of {O β } such that v γ ∈ A if V γ ∩ A = ∅ . Let K = | N ( V ) | be the nerve of V with the canonical G -action.For any x ∈ M , we let ν ( x ) = X i d ( x, M − V i )where d denotes the metric on M which is G -invariant since G acts iso-metrically. Let { v i } be the vertices of | N ( V ) | . Now { P i d ( x,M − V i ) ν ( x ) } is a G -partition of unity subordinate to V . Define the G -map ϕ : M → | N ( V ) | by ϕ ( x ) = X i d ( x, M − V i ) ν ( x ) v i . Note that ϕ is a barycentric mapping. Consider the G -map ψ = r ◦ η : | N ( V ) | → M , where η is the map ψ as in the proof of Proposition 2.3 of[20]. It is straightforward to check that the G -maps ϕ and ψ yield the desired G -domination. Note that K is of dimension less than or equal to n since the V is a refinement. It follows that K must be of dimension n otherwise K has no homology in dimension n whereas dim M = n and M is a compactmanifold of dimension n . Finally, we let V A = { ( V i ∩ A, v i ) } . It followsthat V A is a G -covering of A and the nerve L = | N ( V A ) | is a subcomplex of K . Since A is a compact manifold of dimension k , we conclude that L is ofdimension k and that L equivariantly dominates A . (cid:3) Remark . It has been noted by S. Antonyan in [2] that the equivariantembedding theorem [21, Theorem 6.2] of M. Murayama is incorrect: inthat the Banach space B ( M ) of all bounded continuous functions on M used in [21] is not a Banach G -space and the G -action defined there is notcontinuous. Likewise, the same mistake was also committed by S. Kwasikin [20]. Nevertheless, the G -domination theorem in both [20] and [21] isstated correctly and their proofs are valid provided one replaces B ( M ) with LUC´ILIA BORSARI, FERNANDA CARDONA, AND PETER WONG the linear subspace A ( M ) of all G -uniform functions as in [2]. We thankM. Golasi´nski for bringing [2] to our attention. As noted by Hanner in [12],in non-equivariant settings Borsuk showed in [4] that any compact ANRis dominated by a finite polyhedron. Then, in [6], Brooks showed, againin the non-equivariant setting, that if an n -dimensional compact ANR isdominated by a complex then it is dominated by its n -dimensional skeleton.In order to prove the next proposition, we will need the following non-equivariant result. Lemma 3.2.
Let M be a n -manifold and A ⊂ M a submanifold. Consider c an n -cell in M − A , with ∂c ⊂ A and let σ : ∂c −→ T A be a path field and o be a point in the interior of c . Then, there exists a path field σ ′ : c −→ T M ,extending σ with o being its only singularity in the interior of c . Moreover,in case σ has singularities then we may take σ ′ without singularities in theinterior of c .Proof. Let c ′ be an n -cell contained in Int c , with o in its interior. We willextend σ to c − Int c ′ without creating new singularities: Let [ o , b x ] be theoriented segment through x , beginning at o , ending at b x ∈ ∂c . Thereforewe could write any x ∈ c − Int c ′ as x = (1 − t x ) o + t x b x , where t x ∈ ]0 , b x cc’o The extended path field σ ′ will be defined for each x ∈ c − Int c ′ as follows: σ ′ ( x )( s ) = (1 − t x − s ) o + ( t x + s ) b x , ≤ s ≤ − t x σ ( b x ) (cid:18) s + t x − t x (cid:19) , − t x ≤ s ≤ QUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS 7
Observe that it is well defined because t x >
0, for any x ∈ c − Int c ′ . Also,in the first equation, for any t x when s = 0 we have (1 − t x ) o + t x b x = x ;when s = 1 − t x we have ( t x + 1 − t x ) b x = b x . In the second equation,when s = 1 − t x we have σ ( b x )( − t x + t x − t x ) = σ ( b x )(0) = b x ; when s = 1 wehave σ ( b x )( t x − t x ) = σ ( b x )(1). Therefore σ ′ has no other singularities thanthose of σ (if it has any, they will be in the boundary of c ), so σ ′ has nosingularities in the boundary of c ′ . By Lemma 1.5 of [7] it can be extendedto Int c ′ having only o as a singularity in Int c ′ .By an abuse of notation we will denote this extension of σ ′ to Int c ′ alsoby σ ′ . Therefore, we constructed an extension of σ , σ ′ : c −→ T M , which hasonly one singularity in Int c and in the boundary only the singularities that σ had.If σ does have singularities in the boundary of the cell, we will eliminatethe singularity of σ ′ in its interior:Let y ∈ ∂c be a singularity of σ (therefore a singularity of σ ′ ); let c ⊂ c ⊂ c be two cells such that ∂c ∩ ∂c = { y } , ∂c i ∩ ∂c = { y } , for i = 1 , o ∈ Int c (and therefore o ∈ Int c ).Let [ b x , y ] be the oriented segment through x , beginning at b x ∈ ∂c ,ending at the singularity y ∈ ∂c . Therefore we could write any x ∈ c as x = (1 − t x ) b x + t x y , where t x ∈ [0 , a x ∈ ∂c such that a x = (1 − t x ) b x + t x y , with t x ∈ ]0 , a x b x c c y o c The new path field σ will be defined in each one of the regions representedabove, as follows: LUC´ILIA BORSARI, FERNANDA CARDONA, AND PETER WONG – For x ∈ Int c , we have 0 < t x < σ ( x )( s ) = (1 − t x − s ) b x + ( t x + s ) y, ≤ s ≤ − t x σ ′ ( y ) (cid:18) s + t x − t x (cid:19) , − t x ≤ s ≤ t x >
0, for any x ∈ Int c . Also,in the first equation, for any t x when s = 0 we have (1 − t x ) b x + t x y = x ;when s = 1 − t x we have ( t x + 1 − t x ) y = y . In the second equation, when s = 1 − t x we have σ ′ ( y )( − t x + t x − t x ) = σ ′ ( y )(0) = y .– For x ∈ c − Int c , we have 0 ≤ t x ≤ t x and σ ( x )( s ) = (1 − t x − s ) b x + ( t x + s ) y, ≤ s ≤ t x t x − t x σ ′ (cid:18)(cid:18) − t x t x (cid:19) b x + t x t x y (cid:19) s − ( t x t x − t x )1 − ( t x t x − t x ) ! , t x t x − t x ≤ s ≤ t x >
0, for any x ∈ c . Also, inthe first equation, for any t x when s = 0 we have (1 − t x ) b x + t x y = x ; when s = t x t x − t x we have (1 − t x − ( t x t x − t x )) b x + ( t x + t x t x − t x ) y = (1 − t x t x ) b x + t x t x y .In the second equation, when s = t x t x − t x we have σ ′ ((1 − t x t x ) b x + t x t x y )(0) =(1 − t x t x ) b x + t x t x y .– For x ∈ c − Int c , σ ( x ) = σ ′ ( x ) . Notice that if x ∈ ∂c then t x = t x and if x ∈ ∂c then t x = 0 andtherefore σ is well-defined and continuous in ∂c and ∂c , the boundaries of c and c . A simple verification will show that σ has no singularities in Int c and the fact that σ is a path field in A guarantees that σ is in fact a pathfield. (cid:3) Proposition 3.3.
Let M be a locally smooth G − manifold, dim M = n , A ⊂ M an invariant submanifold so that G acts freely on M − A . Given anequivariant section σ A : A −→ T A , with a finite number of singular orbits, itis possible to extend σ A to an equivariant section σ : M −→ T M in such away the closure of each component of M − A intersects at most one singularorbit of σ . QUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS 9
Proof.
Consider the following diagram: i * q L ϕ * σ M q M ) ϕ * T( M ( ) /L ϕ * K q σ KL T ϕ /L )()( *L σψ /L ϕ /L q A σ A T A T M L ϕψ K i MA
Here, K and L are as in Theorem 3.1 and q L : ( ϕ | L ) ∗ ( T L ) −→ L and q K : ϕ ∗ ( T M ) −→ K denote the pullbacks of q A and q M by ϕ , respectively.Now, starting with σ A , a G -section of q A : T A → A , we define a G -section σ L : L → ( ϕ | L ) ∗ ( T A ), by σ L ( y ) = ( y, σ A ( ϕ ( y )). A similar procedure asthe one indicated in Lemma 1.6 in [7] can be used to extend σ L to a G -section σ K , having only a finite number of singular orbits in K − L . Inorder to extend σ L to an m -simplex δ of K − L , we use Lemma 3.2 andextend it to gδ in the usual equivariant way. Define σ ′ M : M → T M by σ ′ M ( x ) = ϕ ( σ K ( ψ ( x )). Then σ ′ M is an equivariant map, but it is not asection. In fact, it is a homotopy section since q M ( σ M ( x )) = ϕ ◦ ψ ( x ).Consider h : M × I → M , the G -homotopy between ϕ ◦ ψ and the identityon M , and f : A × I ∪ M × → T M given by the G -homotopy between σ A and ( σ ′ M ) | A on A × I and by σ ′ M in M ×
0. Since M admits an invariantmetric, we may apply Lemma 2.1 to obtain an equivariant lifting f ′ of h extending f . Define σ M : M → T M by σ M ( x ) = f ′ ( x, σ M is a G -section on M extending σ A .The first step is to change σ M to reduce the singular set in M − A toa finite one. In order to do this, consider first { Gx , Gx , ..., Gx r } the set of singular orbits of σ K in K − L . The set of singular orbits of σ M , whichare not in A , lies in the pre-image of { Gx , Gx , ..., Gx r } under ψ . Since ψ is equivariant, this set is { Gψ − ( x ) , ..., Gψ − ( x r ) } . Since G acts freely in M − A , for each i , the sets gψ − ( x i ), g in G , are disjoint. Following the proofof Theorem 1.10 of [7], we can assume that for a connected component C of M − A , ψ − ( x ) ∩ C is contained in the interior of c , a closed topological n -cell (see the figure below). d (x −1 ψ ) (x ) g ψ −1 (x ) g ψ −1 (x −1 ψ ) d d c gc Now, consider in M an invariant metric and let d be the distance between ψ − ( x ) and S i =1 Gψ − ( x i ). Let d be the distance between ψ − ( x ) andthe boundary of c and d the distance between ψ − ( x ) and ( Gψ − ( x ) − ψ − ( x )) ∩ C . Finally, take d to be the minimum of { d , d , d } . If weconsider a finite triangulation of c with mesh size less than d/
3, then noclosed simplex of c intersecting ψ − ( x ) intersects a simplex which touches[ S i ( Gψ − ( x i ) − ψ − ( x )) S ∂c ] ∩ C = R C . Let P C be the subpolyhedronof c consisting of simplices which do not intersect R C and let Q C be thesubpolyhedron of P C consisting of those simplices which do not intersect ψ − ( x ). Then σ M | Q C has no singularities and again, by the same procedureused in Lemma 1.6 in [7], we may extend it to P C with a finite number ofsingularities, say, { y , y , ..., y m } . Since the metric on M is invariant, we may QUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS 11 triangulate g i c in the same way we triangulate c so that the complexes g i P C and g i Q C will be the complexes corresponding to P C and Q C for g i ψ − ( x )in g i Q C . By doing so, the singularities of the extension of σ M | g i Q C willbe { g i y , g i y , ..., g i y m } . Finally, we extend this section to M by making itagree with σ M outside G C P C , where G C = { g ∈ G | gC = C } . Repeatingthis procedure for i >
1, we end up with an equivariant section extending σ A with a finite number of singular orbits, { Gy , Gy , ..., Gy m } , lying in theclosure of various components of M − A .The next step is to reduce the set of singularities in such a way the closureof each component of M − A meet at most one singular orbit. For this, let C be a component of M − A and G C y , ...G C y r be all singularities in C .Consider e a closed n -cell in C containing this entire set of singularities inits interior. Let e be another closed n -cell contained in the interior of e such that y , y , ..., y r are in e and g l e T g j e = ∅ , for g l and g j in G C , l = j , as in the figure below. e y gy gegy y e Applying once more Lemma 3.2 for the n -cell e we may reduce the set { y , y , ..., y m } to a single singularity, say z . Doing the same for the cells g j e ,we end up with a cross section τ : M −→ T M with only G C z as singularitiesin C . Repeating this procedure for all other components we are able toextend the path field to M in such a way that the closure of each component C of M − A meet at most two singular orbits, one lying in the interior of C and the other in its boundary. Now, for a component C of M − A let G C z ∪ G C w be its set of singularities,where z ∈ C and w ∈ ( C ) − C . Then, it is possible to find | G C | cells touching( C ) − C only in gw , g ∈ G C and so that each of them contains only one pairof points of the form gz , gw , g ∈ G C , as in the figure below. (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) zw cgz gw Then applying Lemma 3.2 we see that we may change σ in the interiorof the cell containing z and w so that the only singularity left is w . Thenrepeating the procedure to the other cells, equivariantly, we complete theproof. (cid:3) Proposition 3.4.
Let M be a G − manifold, A ⊂ M n an invariant subman-ifold so that G acts freely on M − A . Assume that A admits a G -path fieldwithout singularities. Then M admits a G -path field with no singular orbitsiff | χ | ( M − A ) = 0 .Proof. Assume first | χ | ( M − A ) = 0 and let σ be a G -path field on M with a single singular orbit, say Gx , x in a component C of M − A . Take f : M −→ M , f ( x ) = σ ( x )(1). Then f has only one fixed orbit in C , namely G C x . Since G is finite and χ ( C ) = 0 we have that the sum of the fixedpoint indices of f at gx , g in G C , must vanish. Since the action of G on M is locally smooth and the fixed points are isolated and lie in the sameorbit, it is not hard to see that they have all the same index, and thereforeindex zero. Because the action is free in M − A , we can find an Euclideanneighborhood, U , of x in C such that gU ∩ hU = ∅ , for all g and h in G C .Applying exactly the same procedure as in the proof of Theorem 2.3 of [7],we conclude that it is possible to construct a path field σ ′ over M so that itagrees with σ in M − U and has no singularities in U . Define τ : M −→ T M QUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS 13 to agree with σ in M − ∪ g ∈ G C gU and, for y ∈ gU , τ ( y ) = gσ ′ ( g − y ). It isnot difficult to see that τ is a G -path field over M without singular orbitsin C . The proof is complete if we repeat the same procedure for all othercomponents of M − A .Now, suppose σ A has no singularities and can be extended to M . Let { C , C , ..., C r } and { A , A , ..., A l } be the connected components of M − A and A , respectively. Since, σ | A j : A j → A j has no singularities, we have that χ ( A j ) = 0, for all j . Consider D i the union of the components of A that meetthe closure of C i . Set e C i = D i ∪ C i and let U i be a tubular neighborhood of D i in e C i . Then χ ( D i ) = 0 and χ ( U i − D i ) = 0, since U i − D i fibers over D i . Now, χ ( C i ) = χ ( e C i ) − χ ( U i ) + χ ( U i ∩ C i ) = χ ( e C i ) − χ ( D i ) + χ ( U i − D i ) = χ ( e C i ).Using the compactness of e C i and the fact that σ ( x ) starts at x , it ispossible to find t i ∈ [0 ,
1] so that σ ( x )([0 , t i ]) ⊂ e C i , for all x in e C i . Thereforethe map f i : e C i → e C i given by f i ( x ) = σ ( x )( t i ) is fixed point free, and thisimplies that χ ( e C i ) = 0. So we conclude that χ ( C i ) = 0. This completes theproof. (cid:3) Proposition 3.5.
Suppose G acts on a manifold M with only one orbittype ( H ) . Then it is possible to construct a G -path field on M with a singlesingular orbit. Moreover, M admits a G -path field with no singular orbitsiff | χ | ( M H ) = 0 .Proof. Since
N H/H acts freely on M H then, by Proposition 3.3, it is possi-ble to construct a N H/H -path field σ ′ : M H −→ M H with only one N H/H -singular orbit. Since G acts on M with only one orbit type ( H ), we havethat M is G -homeomorphic to G × NH M H , where N H is the normalizer of H (see [5]). Define σ : M −→ T M by, σ [ g, x ] = gσ ′ ( x ). It is not hard to seethat σ is an equivariant section with a single singular orbit.Now, assume | χ | ( M H ) = 0. Then | χ | ( M H ) = 0 and the section σ ′ can betaken without singularities and so does σ . Finally, if M admits a G -pathfield with no singular orbits, then f : M −→ M given by f ( x ) = σ ( x )(1),has no fixed orbits. Therefore f H : M H −→ M H has no fixed points, whichimplies that | χ | ( M H ) = 0 and the proof is done. (cid:3) Theorem 3.6.
Let G be a finite group and M a compact locally smooth G -space. Then there exists a G -path field on M having at most one singular orbit in the closure of each component of M H . Moreover, M admits a nonsingular G -path field iff | χ | ( M H ) = 0 , for all H ≤ G .Proof. Consider ( H ) , ( H ) , ... ( H r ) the orbit types of the G -action on M ordered in a way that ( H i ) ⊂ ( H j ) implies j ≤ i . For each i ∈ { , , ...r } ,let M i = { x ∈ M | ( G x ) = ( H j ) , j ≤ i } . Then M ⊂ M ⊂ ... ⊂ M r , M = M ( H ) , M r = M and M i − M i − = M ( H i ) . Here, M ( H j ) = { x ∈ M | ( G x ) = ( H j ) } .To prove the first part we will use induction on r . If r = 1, then M hasonly one orbit type, namely, ( H ). Therefore, Proposition 3.5 implies that M admits a G -path field σ with only one singular orbit.Suppose we have succeeded extending σ to a G -path field, σ i − , on M i − so that the closure of each component of M i − − M i − = M H i − intersectsat most one singular orbit. Take N = M H i − ( M i − ∩ M H i ) = M H i . Since N H i /H i acts freely on N , Proposition 3.5 implies that we are able to ex-tend σ i − | M i − ∩ M Hi to an N H i /H i -path field, ¯ σ i : M H i → T M Hi , without N H i /H i -singular orbits.Define e σ i : M ( H i ) → T M ( Hi ) by e σ i ( x ) = l ¯ σ i ( l − x ), where l ∈ G issuch that G x ⊃ lH i l − . Then, e σ i is a well defined G -path field extend-ing σ i − | M i − ∩ M Hi .Now let σ i : M i → T M i coincide with e σ i , on M ( H i ) and with σ i − in M i − .It is not difficult to see that σ i is a G -path field extending σ i − with thedesired property.For the second part, assume | χ | ( M H ) = 0, ∀ H ≤ G . The G -path field σ constructed above can be taken without singular orbits by making use ofProposition 3.4, inductively on { ( H ) } .Finally, if M admits a non-singular G -path field σ then looking at M i − ∩ M H i as an N H i -submanifold of M H i we may repeat the proof of Proposi-tion 3.4, to obtain that | χ | ( M H i ) = 0. The proof is complete. (cid:3) G -Complete Invariance Property In this section, we study related problems concerning the fixed pointtheory for G -deformations. Recall that a G -space X is said to have the G -CIP for G -deformations ( G -CIPD), if for any nonempty closed invariantsubset A ⊆ X , there exists a G -deformation λ ∼ G X such that F ix λ = QUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS 15 A . In [27], necessary and sufficient conditions were given for smooth G -manifolds to possess the G -CIPD. As an application of Theorem 3.6, weobtain the following Theorem 4.1.
Let G be a finite group and M a compact locally smooth G -manifold. Suppose for each isotropy type ( H ) , M H has dimension at least . Let A ⊂ M be a non-empty closed invariant subset. Then the followingare equivalent: • There exists a G -deformation ϕ : M → M such that A = F ix ϕ . • A ∩ ¯ C = ∅ whenever χ ( C ) = 0 for any connected component C of M H and ¯ C denotes the closure of C in M H .Proof. Suppose that there exists a G -deformation ϕ : M → M such that F ix ϕ = A . Let C be a connected component of M H such that χ ( C ) =0. By excision, we have H ∗ ( M H , M H − C ) = H ∗ ( M H , M H − C ) so that χ ( M H , M H − C ) = χ ( M H , M H − C ) = χ ( C ) = 0. Using the relativeLefschetz fixed point theorem and Proposition 2.1 of [25], we conclude that ϕ must have a fixed point in the closure ¯ C of C in M H . Since F ix ϕ = A ,it follows that A ∩ ¯ C = ∅ .Conversely, if χ ( C ) = 0 then by Theorem 3.6 there exists a G -path field σ such that σ has one singular orbit in G ¯ C . If A ∩ C = ∅ , this singular orbitlies in A ∩ GC . If A ∩ C = ∅ , then the singular orbit must lie in M > ( H ) since M H is open and dense in M H and M H = M H ∪ ( M H ∩ M > ( H ) ), where M > ( H ) denotes the set of points of M of isotropy type ( K ) > ( H ). Now, let ϕ ( x ) = σ ( x )( t x ) where t x = d ( x, A ) and d is a bounded G -invariant metric.Then, we have F ix ϕ = A and ϕ ∼ M . (cid:3) Remark . In the case where M is a smooth G -manifold and M H /W H is connected for each ( H ), the necessary and sufficient conditions obtainedin [27] can be derived from those of Theorem 4.1. Our formulation resem-bles closely to case (B) of Theorem 1 of [16] except that Jiang consideredconnected components C of M H instead. According to case (B) of [16, The-orem 1], there is a fixed point free G -deformation if M H is connected and χ ( M H ) = 0 for all ( H ). However, counter-examples have been found byD. Ferrario [11]. Therefore, Theorem 4.1 gives the correct necessary andsufficient conditions. Remark . The first example of a G -space X in which each of the identitymaps 1 X H : X H → X H is deformable to be fixed point free but 1 X is not G -deformable to be fixed point free was given by M. Izydorek and A. Vidal[15]. We would like to point out that one can easily modify their example(by taking the cartesion product with the unit interval) to give an exampleof a G -Wecken complex in the sense of [26] such that the equivariant Eulercharacteristic used in [26] is nonzero, that is, the identity is not equivariantlydeformable to be fixed point free. The case for smooth G -manifolds wasstudied by D. Ferrario in [11] for more general G -maps. References [1] G.Allaud and E. Fadell, A fiber homotopy extension theorem,
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Bull. Amer. Math. Soc. (1973),133–135.[25] D. Wilczy´nski, Fixed point free equivariant homotopy classes Fund. Math. (1984),47–60.[26] P. Wong, Equivariant path fields on G -complexes, Rocky Mountain J. Math. (1992), 1139–1145.[27] P. Wong, On the location of fixed points of G -deformations, Topol. Appl. (1991),159–165. Departamento de Matem´atica - Instituto de Matem´atica e Estat´ıstica -Universidade de S˜ao Paulo, Rua do Mat˜ao, 1010 - CEP 05508-090, S˜ao Paulo -SP, Brasil
E-mail address : [email protected] Departamento de Matem´atica - Instituto de Matem´atica e Estat´ıstica -Universidade de S˜ao Paulo, Rua do Mat˜ao, 1010 - CEP 05508-090, S˜ao Paulo -SP, Brasil
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