Equivariant path fields on topological manifolds

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A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf's result to topological manifolds, replacing vector
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    a  r   X   i  v  :   0   7   0   6 .   4   0   9   7  v   1   [  m  a   t   h .   A   T   ]   2   7   J  u  n   2   0   0   7 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. 1.  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 if and 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 was shown 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  Complete Invariance 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  =  Fixf   :=  { x  ∈  M   | f  ( x ) =  x } . Date  : December 26, 2013.2000  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. 1  2 LUC´ILIA BORSARI, FERNANDA CARDONA, AND PETER WONG 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 our purposes, 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 - Covering Homotopy Property   ( G -CHP) if for all  G -space  X   the following commutativediagram has a solution  F   :  X  × [0 , 1]  →  E   where all maps are  G -equivariant.  EQUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS 3 X   ×{ 0 }  f  −−−−→  E  incl.   p X   × [0 , 1]  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    p  :  E  I  →  Ω  p  bethe  G -map defined by    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   F   : Ω  p  × I   →  B  whichextends  f  . Then  λ  : Ω  p  →  E  I  , defined by  λ ( e,w )( t ) =   F  ( e,w,t ), is anequivariant lifting function for  p , that is,    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   × 0 ∪ A × I   →  E   and   h  :  X   × I   →  M   such that   p ◦ f   =  h | ( X  × 0 ∪ A × I  ) .Then, there exists a   G -map   f   :  X   × I   →  E   which extends   f   and such that   p ◦   f   =  h .Proof.  This lemma is an equivariant version of Theorem 2.4 of  [1]. The proof of 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 of Corollary 2.3 of  [25].   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  0 M   of all paths  α on  M   such that for 0  ≤  t  ≤  1,  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  4 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  0 M  ,q  M  | T  0 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  0 M   :  T  0 M   →  M   is also a  G -subfibration. The notions of non-singular and of simple  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  ) =  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 of dimension   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  EQUIVARIANT 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 ) =  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  {  id ( 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 ) =  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 of dimension  k  and that  L  equivariantly dominates  A .   Remark   3.1 .  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] is stated correctly and their proofs are valid provided one replaces  B ( M  ) with
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