The Burnside Ring-Valued Morse Formula for Vector Fields on Manifolds with Boundary
aa r X i v : . [ m a t h . G T ] J u l THE BURNSIDE RING-VALUED MORSE FORMULA FORVECTOR FIELDS ON MANIFOLDS WITH BOUNDARY
GABRIEL KATZ
Abstract.
Let G be a compact Lie group and A ( G ) its Burnside Ring. Fora compact smooth n -dimensional G -manifold X equipped with a generic G -invariant vector field v , we prove an equivariant analog of the Morse formula Ind G ( v ) = n X k =0 ( − k χ G ( ∂ + k X )which takes its values in A ( G ). Here Ind G ( v ) denotes the equivariant index ofthe field v , { ∂ + k X } the v -induced Morse stratification (see [M]) of the boundary ∂X , and χ G ( ∂ + k X ) the class of the ( n − k )-manifold ∂ + k X in A ( G ).We examine some applications of this formula to the equivariant real alge-braic fields v in compact domains X ⊂ R n defined via a generic polynomialinequality. Next, we link the above formula with the equivariant degrees ofcertain Gauss maps. This link is an equivariant generalization of Gottlieb’sformulas ([G], [G1]). Introduction
Let X be a compact smooth n -manifold with boundary ∂X . A generic (seeDefinition 1.1) vector field v on X which is nonzero along ∂X gives rise to a strat-ification X := ∂ +0 X ⊃ ∂ +1 X ⊃ ∂ +2 X ⊃ · · · ⊃ ∂ + n X (1.1)by compact submanifolds, where dim ( ∂ + j X ) = n − j . Here ∂ +1 X is the part of theboundary ∂ X := ∂X where v points inside X . By definition, ∂ X is the locuswhere v is tangent to the boundary ∂ X . Its portion ∂ +2 X ⊂ ∂ X consists of pointswhere v points inside ∂ +1 X . Similarly, ∂ X is the locus where v is tangent to ∂ X .In the same spirit, ∂ +3 X ⊂ ∂ X consists of points where v points inside ∂ +2 X .Continuing this process, we get the Morse stratification (1.1).In Section 2, Theorem 2.1, we investigate quite strong restrictions on the natureof the stratification { ∂ j X } (and thus of { ∂ + j X } ) imposed by a G -symmetry of thefield v and the manifold X . Definition 1.1.
We say that a field v is generic , if for each k , viewed as a sectionof the bundle T ( ∂ k X ) | ∂ k +1 X , v is transversal to the zero section of the tangentbundle T ( ∂ k +1 X ) . In his groundbreaking 1929 paper [Mo], Morse discovered a beautiful connectionbetween stratification (1.1) and the index
Ind ( v ) of the field v . It is expressed in Date : November 21, 2018. terms of the Euler numbers of the strata from (1 . Ind ( v ) = n X k =0 ( − k χ ( ∂ + k X ) . (1.2)Our main observation is that, for any compact Lie group G and a generic G -equivariant vector field v , a similar formula with values in the Burnside ring A ( G )holds. Theorem 3.1 is an equivariant version of the Morse formula (1 .
2) for genericsymmetric vector fields on manifolds with boundary. It is a generalization of Theo-rem 6.6, [LR], for fields that do not necessarily point outward X along its boundary .In Section 4, Theorem 4.2, we combine some results of Khovanskii [Kh] about(non-equivariant) indices of real algebraic vector fields in polynomially-defined do-mains in R n with formulas (3 .
9) and (3 .
10) from Theorem 3.1 to get a handle onthe size of eqivariant indices of such fields.In Section 5, Theorem 5.1, we obtain an A ( G )-valued version of Gottlieb’s ”Topo-logical Gauss-Bonnet Theorem” [G], [G1]. These results connect the indices ofpullback fields F ∗ w under smooth G -maps F : X → V to the equivariant degree Deg G ( γ ) of the Gauss map γ : ∂X → S ( V ). Here w is a nonsingular G -invariantfield on a space of a G -representation V , and F | ∂X is an immersion.In Section 6, we speculate about some generalizations of our equivariant indexformulas, the generalizations which reside in refined versions of the ring A ( G ). Acknowledgments:
I am grateful to Wolfgang L¨uck for a valuable conversationwhich launched this investigation.2. G -invariant Vector Fields and Morse Stratifications When v is both generic and G -invariant, the Morse stratification { ∂ + k X } is in-variant as well. In fact, it is quite special. For example, the following propositionis valid: Theorem 2.1.
Let a compact Lie group G act faithfully on a smooth compact n -manifold X with an oriented boundary ∂ X . Then, for any k and a generic G -equivariant vector field v , • the main orbit-type of ∂ k X is G . • for all k > n − dim ( G ) , we have ∂ k X = ∅ . • the sets ∂ ± n − dim ( G ) X are disjoint unions of free G -orbits. • if G is connected, the sets ∂ ± n − − dim ( G ) X are disjoint unions of the following G -spaces: (1) G × [0 , , (2) G × S , (3) the mapping cones C ( G → G/H ) , where H ≈ SO (2) or SU (2) , (4) C ( G → G/H ) ∪ G C ( G → G/K ) , where H, K ≈ SO (2) or SU (2) . Actually, [LR] deals with the category of cocompact discrete G -actions. That is, (
X, v ) is ( n + 1 − dim ( G ))-convex in the terminology of [K]. Proof.
Pick x ∈ ∂ k X ◦ := ∂ k X \ ∂ k +1 X and let H = G x . Because v is H -invariant, a linearization of the H -action at x must leave the flag F x of tangent half-spaces { T + x ( ∂ + l X ) } l ≤ k invariant as well. Pick a G -invariant metric on X . Considerthe unique ”frame” ν x comprised of k mutually orthogonal rays r , . . . , r k gener-ated by intersecting the flag F x with the space N kx , normal in T x ( X ) to T x ( ∂ k X ).Specifically, for all l < k , the positive cone Span + { r , . . . , r l } = N kx ∩ T + x ( ∂ + k − l X ).The ”frame” ν x must be H -invariant. Thus, in a basis consistent with ν x , the H -action is diagonal with the positive eigenvalues. Since H is compact, its actionon the space N kx (spanned by ν x ) is trivial. At the same time, the H -action on thespace tangent to the trajectory Gx at x is trivial as well.Now consider the 0-dimensional G -invariant sets ∂ ± n X . By the argument above,for any x ∈ ∂ ± n X , the group H = G x acts trivially on the n -flag F x , and thus, on T x X . Since the G -action on X is faithful, we get H = 1. Hence, either ∂ ± n X = ∅ when dim ( G ) >
0, or dim ( G ) = 0 and ∂ ± n X is a union of free orbits.Next, consider the case dim ( G ) ≥ G -invariantsets ∂ ± n − X . By the argument above, for any x ∈ ∂ ± n − X , the group H = G x acts trivially on the on the vector space N n − x . In addition, H acts trivially onthe 1-dimensional space tangent to the orbit G/H through x . Since ∂ n − X is1-dimensional, the space tangent to the orbit and the space tangent to ∂ n − X coincide. As a result, the H action is trivial on T x X , and thus, in the vicinity of x .Since the G -action on X is faithful, we get again H = 1. Hence, either dim ( G ) > ∂ ± n − X = ∅ , or dim ( G ) = 1 and ∂ ± n − X is a union of free G -orbits.Finally, let us treat the general case. For k > G -invariant v , ∂ k X is a closed G -manifold. Pick a main orbit G/H ⊂ ∂ k X , H = G x for some x ∈ ∂ k X .According to [B], an invariant regular neighborhood of G/H ⊂ ∂ k X is diffeomorphicto the balanced product G × H V , where V is a space of an H -representation ψ . Ifthis ψ is a non-trivial representation, G/H is not the main orbit: the stationarygroups of some points from G × H ( V \ { } ) must be smaller than H . Thus, wecan assume that H acts trivially on V . In addition, it acts trivially on the tangentspace to the orbit G/G x at x , as well as on the normal space N kx . As a result, the G x -action on T x X is trivial. Because G acts faithfully on X , G x = 1; that is, themain orbit-type of ∂ k X (and hence of ∂ ± k X ) is G ! Therefore, when n − k < dim ( G ), ∂ k X = ∅ .When n − k = dim ( G ), each main orbit G is open and dense in some G -invariantunion of connected components of ∂ k X . Since both G and ∂ k X are closed manifolds, ∂ k X is G -diffeomorphic to a disjoint union of several G ’s.When n − k = dim ( G ) + 1 and G is connected, it is present as a codimensionone main orbit-type of each connected component of ∂ k X . Fortunately, connectedclosed G -manifolds of this kind are very rigid: they all are either products G × S , orunions of two mapping cylinders C ( G → G/H ), C ( G → G/K ) with H and K beingdiffeomorphic to a sphere [B], [K1]. The two mapping cylinders share the same”top” G . The only spheres among the Lie groups are O (1) ≈ Z , SO (2) , SU (2).However, the Z -option must be excluded: it leads to a non-orientable manifold,while all ∂ k X are oriented for k >
0. This leaves us with the models C ( G → G/H ) ∪ G C ( G → G/K ) , where H, K = SO (2) , SU (2), for the components of ∂ n − − dim ( G ) X . GABRIEL KATZ
Similarly, the models for ∂ ± n − − dim ( G ) X are: C ( G → G/H ) ∪ G C ( G → G/K ), C ( G → G/H ), G × S , and G × [0 , H, K = SO (2) , SU (2). (cid:3) Corollary 2.1.
Let
X, v be as in Theorem 2.1. Assume that G is finite. Then,for some integer l ≥ , | ∂ n X | = 2 l | G | , provided | G | being odd, and | ∂ n X | = l | G | ,provided | G | being even.If G = SO (2) , then ∂ n X = ∅ , ∂ ± n − X each is a union of circles on which G acts freely, and each ∂ ± n − X is a union of tori SO (2) × S , cylinders SO (2) × [0 , ,2-disks, and 2-spheres. The SO (2) -action on each disk or sphere is semi-free.Question. For a given compact G -manifold X of dimension n with orientableboundary, what is the minimal number of free trajectories that form the sets ∂ n − dim ( G ) ( X ) and ∂ ± n − dim ( G ) ( X )? The minimum is taken over the space of allgeneric G -invariant vector fields v . Lemma 2.1.
For a G -invariant field v and each compact subgroup H ⊂ G , wehave ∂ + k ( X H , v | X H ) = X H ∩ ∂ + k ( X, v ) .Proof. A key observation here is that, for each compact subgroup H ⊂ G , any G -invariant field v on X is tangent to the submanifold X H ⊂ X . Indeed, in a G -invariant metric g , the normal to X H component ν H of v must be H -invariant,and thus vanishes for H = 1. Each submanifold X H meets ∂ X transversally: justconsider a linearization of the G x -action, G x ⊃ H , at a typical point x ∈ X H ∩ ∂ X .Since v is tangent to X H , evidently, ∂ +1 ( X H , v | X H ) = X H ∩ ∂ +1 ( X, v ).We proceed by induction on k . Assume that X H and ∂ s X are transversal, ∂ s ( X H , v | X H ) = X H ∩ ∂ s ( X, v ), and ∂ + s ( X H , v | X H ) = X H ∩ ∂ + s ( X, v ) for all s < k . If x ∈ X H ∩ ∂ + k ( X, v ) and v ( x ) is tangent to ∂ k − X ∩ X H = ∂ k − X H ,then v ( x ) points inside of ∂ + k − X if and only if it points inside ∂ + k − X H . Thus, ∂ k ( X H , v | X H ) = X H ∩ ∂ k ( X, v ), which completes the induction step. (cid:3) Equivariant Morse Formula for Vector Fields
Following tom Dieck, consider a ring A ( G ) generated over Z by equivalenceclasses of compact differentiable G -manifolds. Two G -manifolds X and Y are saidto be equivalent, if for any compact subgroup H ⊂ G , the Euler numbers χ ( X H )and χ ( Y H ) are equal. This equivalence relation respects disjoint unions and carte-sian products of G -spaces. Therefore, A ( G ) is a ring with the sum and productoperations induced by disjoint unions and cartesian product of equivalence classesof G -manifolds.In particular, any compact G -manifold X gives rise to an element χ G ( X ) ∈ A ( G ).Let ch : A ( G ) → Q ( H ) ∈ conj ( G ) Z be a ring homomorphism induced by the cor-respondence χ G ( X ) → { χ ( X H ) } ( H ) ∈ conj ( G ) , where conj ( G ) denotes the conjugacyclasses of compact subgroups in G . It turns out that ch is a monomorphism [D].We denote by ch H the ( H )-indexed component of ch .Consider only the conjugacy classes ( H ) with the finite quotients W H := N H/H ( N H standing for the normalizer of H in G ) and denote by Φ( G ) the set of such ( H ).Next, form a free abelian group A ′ ( G ) generated by elements of Φ( G ) (equivalently,by the orbit-types { G/H } ( H ) ∈ Φ( G ) regarded as G -spaces). By [D], Theorem 1, the In [K] we proved that, for n = 3 and G = 1, this minimum is zero. natural homomorphism A ′ ( G ) → A ( G ) is an isomorphism. As a result, any element[ X ] ∈ A ( G ) is detected by { ch H ([ X ]) } ( H ) ∈ Φ( G ) alone.Let U Z be a G -invariant regular neighborhood of the zero set Z := Z ( v ) = { x ∈ X | v ( x ) = 0 } and a smooth manifold. Assume that v is in general position with respect to ∂ U Z := ∂U Z . Note that, for each H ⊂ G , U HZ and Z H are homotopy equivalent.Inspired by [M], we propose the following Definition 3.1. ind G ( v ) := n X k =0 ( − k χ G ( ∂ + k U Z ) = χ G ( Z ) + n X k =1 ( − k χ G ( ∂ + k U Z )(3.1) Lemma 3.1.
For each ( H ) ∈ conj ( G ) , ch H ( ind G ( v )) = ind ( v | X H ) , the non-eqivariant index of the field v on X H (equivalently, on U HZ ).Proof. By Lemma 2.1, v is tangent to X H ⊃ U HZ and ( ∂ + k U Z ) H = ∂ + k ( U HZ ).Since v is tangent to X H , all the zeros of v | X H are among the zeros of v on X , andno new zeros appear. By the Morse formula (1 .
2) ([M]), ind ( v | U HZ ) = n X k =0 ( − k χ ( ∂ + k ( U HZ )) = n X k =0 ( − k χ ( ∂ + k ( U Z ) H ) . Thus, ch H ( ind G ( v )) = n X k =0 ( − k ch H (( ∂ + k U Z )) = ind ( v | U HZ ) = ind ( v | X H ) . (3.2) (cid:3) For G -invariant vector fields, there is an alternative definition of the equivariantindex. It is more involved and mimics the classical definition of index as the sum oflocal degrees produced by the field v in the vicinity of its zero set. Let us describethis definition along the lines of [L], [LR].The tangent space T ( x, ( T X ) decomposes as a direct sum of the subspaces T x X ⊕ T x X , where the first factor is thought as being tangent to the zero section ǫ : X → T X and the second one to the fiber of the bundle
T X → X . Let α : T x X ⊕ T x X → T ( x, ( T X ) denote this G x -equivariant isomorphism. Consider a G -equivariant vector field v on X , viewed as a section v : X → T X of the tangent G -bundle T X . Assume that v is transversal to the zero section ǫ (due to the”doubling” nature of the G x -space T ( x, ( T X ), this equivariant transversality of v and ǫ is available by a general position argument). If v ( x ) = 0 for some x ∈ X , then T ( x, ( T X ) decomposes a direct sum of Dv ( T x X ) and Dǫ ( T x X ) as well. Here Dv and Dǫ denote the differentials of the respective maps. Consider the G x -equivariantisomorphism D x,v : T x X Dv −→ T ( x, ( T X ) α − −→ T x X ⊕ T x X pr −→ T x X (3.3), where pr stands for the projection on the second factor. D x,v induces a G x -map D cx,v : T x X c → T x X c on the one point compactification T x X c of T x X . For such In particular, Z is an equivariant deformation retract of U Z GABRIEL KATZ a map, its degree,
Deg G x ( D cx,v ) ∈ A ′ ( G x ) ≈ A ( G x ), is defined in terms of theequivariant Lefschetz classes Λ G x ( D cx,v ), Λ G x ( Id T x X c ) by the formula Deg G x ( D cx,v ) = (cid:2) Λ G x ( D cx,v ) − (cid:3)(cid:2) Λ G x ( Id T x X c ) − (cid:3) (3.4)(see [L], [LR]), where 1 ∈ A ( G x ) stands for the class of a point.In Section 4, we will return briefly to the definition of an equivariant degree Deg G ( f ) for a general G -map f : X → Y of two G -manifolds.Meanwhile, let us recall the definition of the equivariant Lefschetz class Λ G ( f ) ∈ A ′ ( G ) of a G -map f : X → X , X being a finite G − CW -complex . In particular,for each ( H ) ∈ Φ( G ), the orbit-space W H \ X H has the structure of a finite CW -complex. Let Λ G ( f ) := X ( H ) ∈ Φ( G ) L Z [ W H ] ( f H , f >H ) · χ G ( G/H )(3.5), where χ G ( G/H ) is the class of the homogeneous space
G/H in A ′ ( G ), and theinteger L Z [ W H ] ( f H , f >H ) = X p ≥ ( − p · tr Z [ W H ] (cid:0) C p (cid:0) f H , f >H ) (cid:1) . (3.6)In (3 . Z -valued trace tr Z [ W H ] of the f -induced automorphism C p (cid:0) f H , f >H ) of the finitely generated free Z [ W H ]-module C p (cid:0) X H , X >H ; Z ), themodule of the relative celluar p -chains of the pair ( X H , X >H ) (see [LR] for details). Lemma 3.2.
The equivariant Lefschetz class Λ G ( f ) from (3 . is detected by thenon-equivariant Lefschetz numbers { Λ( f H ) ∈ Z } ( H ) ∈ Φ( G ) .Proof. By [D], Theorem 2, only the orbit-types
G/H with
W H being finitecontribute to the homomorphism ch : A ( G ) → Q ( H ) Z . Therefore, we will consideronly the orbit-types from Φ( G ).In (3.5), we took into account only the orbit-types ( H ) ∈ Φ( G ) with non-vanishing coefficients L Z [ W H ] ( f H , f >H ) (defined by (3 . orbit-type ( K ) and apply ch K to Λ G ( f ). Then, ch K (Λ G ( f )) = X ( H ) ∈ Φ( G ) n X p ≥ ( − p tr Z [ W H ] (cid:0) C p ( f H , f >H ) (cid:1)o · χ ( G/H K )= n X p ≥ ( − p tr Z [ W K ] (cid:0) C p ( f K , f >K ) (cid:1)o · χ ( G/K K ) == n X p ≥ ( − p tr Z [ W K ] (cid:0) C p ( f K , f >K ) (cid:1)o · | W K | == X p ≥ ( − p tr Z (cid:0) C p ( f K , f >K ) (cid:1) := Λ( f K ) . (3.7)In other words, for such a minimal ( K ) ∈ Φ( G ), L Z [ W K ] ( f K , f >K ) = | W K | − Λ( f K ) = | W K | − ch K (Λ G ( f )) . for example, see [O] for the definition of a G - CW -complex i.e. a maximal subgroup K ⊂ G The rest of the argument is performed inductively, the induction step beingapplied to a minimal orbit-type ( K ′ ) in Φ( G ) \ ( K ) with a non-zero coefficient L Z [ W K ′ ] ( f K ′ , f >K ′ ). Indeed, compute ch K ′ of β := Λ G ( f ) − L Z [ W K ] ( f K , f >K ) · χ G ( G/K )to conclude, in a similar way, that ch K ′ ( β ) equals Λ( f K ′ ). (cid:3) Corollary 3.1.
Deg G x ( D cx,v ) ∈ A ( G x ) is determined by the Z -valued degrees (cid:8) Deg ( D H,cx,v ) (cid:9) ( H ) ∈ Φ( G x ) of the maps D H,cx,v := D cx,v | : ( T x X c ) H → ( T x X c ) H . For all H / ∈ Φ( G x ) , H ⊂ G x , ch H [ Deg G x ( D cx,v )] = 0 .Proof. Apply the arguments (3.7) of Lemma 3.2 (with G being replaced by G x )to formula (3 . (cid:3) Now put
Ind G ( v ) := X Gx ∈ G \ Z ( v ) G × G x [ Deg G x ( D cx,v )](3.8) Lemma 3.3.
Ind G ( v ) , defined by (3 . , and ind G ( v ) , defined by (3 . , are equal.Proof. For each x from the orbit-space G \ Z ( v ), consider a small G x -equivariant n -disk D x ⊂ X centered on x . Note that, when Z ( v ) is discrete, the quotient G/G x must be finite. Evidently, ∂ + k U Z ≈ ` x G × G x ( ∂ + k D x ). Therefore, in order to prove Ind G ( v ) = ind G ( v ), it will suffice to verify that, for each x ∈ G \ Z ( v ), Deg G x ( D cx,v ) = X k ( − k χ G x ( ∂ + k D x )in A ′ ( G x ). The last equality follows from Corollary 3.1 together with the validity ofnon-equvariant Morse formulas of the type (1 .
2) (with G being replaced by H ⊂ G x and X by D x ). (cid:3) Theorem 3.1.
Let G be a compact Lie group, and let X be a compact smooth n -dimensional G -manifold with orientable boundary ∂ X . Let v be a generic G -invariant vector field on X (with isolated singularities). Then the equivariant Morseformula Ind G ( v ) = ind G ( v ) = n X k =0 ( − k χ G ( ∂ + k X )(3.9) is valid in the ring A ( G ) . If the G -action on X is faithful and dim ( G ) > , Ind G ( v ) = ind G ( v ) = n − − dim ( G ) X k =0 ( − k χ G ( ∂ + k X )(3.10) When G is connected, the contribution of the stratum ∂ + n − − dim ( G ) X to Ind G ( v ) isquite special: χ G ( ∂ + n − − dim ( G ) X ) = X { ( H ) ∈ Φ( G ) | H ≈ SO (2) ,SU (2) } n H · χ G ( G/H ) , also denoted by ∂ +0 X GABRIEL KATZ where { n H } are some nonnegative integers. In particular, if a connected G is suchthat, for each H ⊂ G isomorphic to SO (2) or SU (2) , the group W H is infinite,then the contribution of the stratum ∂ + n − − dim ( G ) X vanishes.Proof. By Lemma 3.3,
Ind G ( v ) = ind G ( v ). Since the elements of A ( G ) aredetected by the character map ch : A ( G ) → Q ( H ) ∈ Φ( G ) Z , it will suffice to check thevalidity of (3.9) by applying ch to both sides of the conjectured equation ind G ( v ) = P nk =0 ( − k χ G ( ∂ + k X ) .By Lemma 2.1, for any H ⊂ G , ch H [ χ G ( ∂ + k X )] := χ (( ∂ + k X ) H ) = χ ( ∂ + k ( X H )).Thus, ch H (cid:0) n X k =0 ( − k χ G ( ∂ + k X ) (cid:1) = n X k =0 ( − k χ ( ∂ + k ( X H ))(3.11)On the other hand, by Lemma 3.1 and formula (3 . ch H ( ind G ( v )) = ind ( v | X H ).Since the Morse formula claims that ind ( v | X H ) = P nk =0 ( − k χ ( ∂ + k ( X H )), we getthe desired equality ch H ( ind G ( v )) = ch H ( P nk =0 ( − k χ G ( ∂ + k X )) for all ( H ) ∈ conj ( G ).In order to derive formula (3 .
10) from formula (3 . ∂ + k X = ∅ for all k > n − dim ( G ). Moreover, ∂ + n − dim ( G ) X is the unionof free G -orbits, and thus χ G ( ∂ + n − dim ( G ) X ) = 0, provided dim ( G ) > A ( G ), model (3) is G -homotopy equivalent to G/H , and model (4), by the additivity of Euler charac-teristics, produces the element χ G ( G/H ) + χ G ( G/K ).If a connected G is such that, for each H ⊂ G isomorphic SO (2) or SU (2), W H is infinite, the space
G/H admits a free S action, S ⊂ W H . Thus, χ G ( G/H ) = 0.As a result, for such a G , we get a simplification of (3 . Ind G ( v ) = n − − dim ( G ) X k =0 ( − k χ G ( ∂ + k X ) . (3.12) (cid:3) Corollary 3.2.
Let X and v be as in Theorem 3.1. Denote by v +1 an orthogonalprojection (with respect to a G -invariant metric on X ) of the field v | ∂ +1 X on thetangent space T ( ∂ +1 X ) . Assume that v +1 has only isolated singularities. Then thefollowing formula holds in A ( G ) : χ G ( X ) = ind G ( v ) + ind G ( v +1 )(3.13) Proof.
In view of the recursive nature of the Morse stratification { ∂ + k X } , (3 . . (cid:3) Polynomial Vector Fields in Polynomial Domains
Now, we turn our attention to polynomial vector fields v in domains in R n thatare defined by polynomial inequalities.Consider a real n -dimensional vector space W and a G -representation Ψ : G → GL R ( W ). Denote by P Ψ the algebra of invariant polynomials on W . Let v be a G -invariant vector field in W ≈ R n which has polynomial components { P i ∈P Ψ } ≤ i ≤ n . Assume that deg ( P i ) ≤ m i . We denote by ˜ P i the homogenized versionsof P i , that is, ˜ P i (1 , x , . . . , x n ) = P i ( x , . . . , x n ).Also consider an invariant polynomial Q ∈ P Ψ of degree d . We say that thepair ( v, Q ) is non-degenerated , if 1) all zeros of v are simple, 2) the system { x =0 , ˜ P i ( x , x , . . . , x n ) = 0 } ≤ i ≤ n has only a trivial solution , and 3) the hypersurface Q = 0 does not contain the zeros of v .For any non-increasing sequence m , m , . . . , m n of natural numbers, form theparallelepiped Π( m , . . . , m n ) in R n defined by { ≤ x i ≤ m i − } ≤ i ≤ n (4.1)Let O ( d, m , . . . , m n ) be the number of integral lattice points ( x , . . . , x n ) inΠ( m , . . . m n ), subject to the inequalities12 ( m + · · · + m n − d − n ) ≤ x + · · · + x n ≤
12 ( m + · · · + m n − n )(4.2)Consider the domain X Q := { w ∈ W | Q ( w ) ≥ } . Then, according to a theoremof Khovanskii (Theorem 1, [Kh]), for a non-degenerated pair ( v, Q ) as above, theabsolute value of index, | ind ( v ) | , in X Q is bounded from above by O ( d, m , . . . , m n );moreover, this estimate is sharp.Assume that X Q is compact with a smooth boundary ∂ X Q and a polynomialfield v is generic (see Definition 1.1) in relation to the boundary. Combining theKhovanskii Theorem and Morse Formula (1 . Theorem 4.1.
For a polynomial field v in X Q as above, (cid:12)(cid:12)(cid:12) n X k =0 ( − k χ ( ∂ + k X Q ) (cid:12)(cid:12)(cid:12) ≤ O ( d, m , . . . , m n )(4.3) Moreover, there exist ( Q, v ) for which the inequality (4 . can be replaced by theequality. Let V ⊂ R n be a vector subspace of dimension l . Let d V ≤ d denote the degree ofthe polynomial Q being restricted to the subspace V . Consider a subspace U ⊂ R n which is spanned by some set of l basic vectors { e j , . . . e j l } in R n and such that theobvious orthogonal projection p U : R n → U , being restricted to V , is onto. Denoteby U ( V ) the finite set of such U ’s.Put O ( V ; d V , m , . . . , m n ) = min U ∈U ( V ) n O ( d V , m j , . . . , m j l ) o , (4.4)where U = span { e j , . . . , e j l } and O ( d V , m j , . . . , m j l ) is defined as in (4 . .
3) for an equivariant setting.
Theorem 4.2.
Let G be a compact Lie group. Pick a sequence m ≥ m ≥ · · · ≥ m n > of integers. Consider a representation Ψ : G → GL ( n, R ) , an invariantpolynomial Q ( x , . . . , x n ) of degree d , and a non-degenerate Ψ( G ) -invariant poly-nomial field v = ( P ( x , . . . , x n ) , . . . , P n ( x , . . . , x n )) i.e. no zero of v escapes to infinity. such that deg ( P i ) ≤ m i . Assume that X Q := { ~x ∈ R n | Q ( ~x ) ≥ } is a compact domain with a smooth boundary ∂ X Q and that v is in general position with respectto ∂ X Q .Then the image Q ( H ) ∈ Φ( G ) z ( H ) of the element P nk =0 ( − k χ ( ∂ + k X Q ) ∈ A ( G ) under the character monomorphism ch : A ( G ) → Q ( H ) ∈ Φ( G ) Z belongs to the par-allelepiped P Φ( G ) defined by the inequalities: (cid:12)(cid:12) z ( H ) (cid:12)(cid:12) ≤ O (( R n ) H ; d ( R n ) H , m , . . . , m n )(4.5) Proof.
Let l = dim (( R n ) H ). For each U = span { e j , . . . , e j l } as above withthe property p U : ( R n ) H → U being onto, the projection p U induces an invertiblelinear transformation of the pairs ( v, X HQ ) and ( p U ( v ) , p U ( X HQ )), where X HQ := X Q ∩ ( R n ) H . Let I be the set of indices complementary to the set J := { j , . . . , j l } .Then the components ˜ P j of p U ( v ) are obtained by substituting { x i = 0 } i ∈ I into { P j ( x , . . . , x n ) } j ∈ J . Note that deg ( ˜ P j ) ≤ deg ( P j ) ≤ m j . Also, p U ( X HQ ) ⊂ Span { e j } j ∈ J is defined by a polynomial inequality ˜ Q ( x j , . . . , x j l ) ≥ Q ( x , . . . , x n ) ≥ x i , i ∈ I, as linear combination of the { x j } j ∈ J . Evidently, deg ( ˜ Q ) = d ( R n ) H ≤ d . Since v | ( R n ) H is parallel to ( R n ) H , p U also maps ∂ + k ( X HQ ) onto ∂ + k [ p U ( X HQ )]. By [Kh] andTheorem 3.1, | Ind ( p U ( v )) | in p U ( X HQ ) (equivalently, | P nk =0 ( − k χ ( ∂ + k p U ( X Q )) | )has an upper boundary O ( d ( R n ) H , m j , . . . , m j l ). Via the linear diffeomorphism p U , Ind ( p U ( v )) in { ˜ Q ≥ } and Ind ( v | X HQ ) in X HQ are equal. Thus, | Ind ( p U ( v )) | ≤ O ( d ( R n ) H , m j , . . . , m j l ). In view of formula (4 .
3) and Lemma 3.1, | ch H ( Ind G ( v )) | = | Ind ( v | X HQ ) | ≤ O ( d ( R n ) H , m j , . . . , m j l ) , and thus using formula-definition (4 . | ch H ( Ind G ( v )) | ≤ O (( R n ) H ; d ( R n ) H , m , . . . , m n ) . (cid:3) Corollary 4.1.
Let v be as in Theorem 4.2 and Ψ be an orthogonal represen-tation. Denote by B r ⊂ R n the ball of radius r centered at the origin. Put d H = dim (( R n ) H ) . Then, for each ( H ) ∈ Φ( G ) and generic r , (cid:12)(cid:12) ch H ( Ind G ( v | B r )) (cid:12)(cid:12) = (cid:12)(cid:12) d H X k =0 ( − k χ ( ∂ + k B Hr ) (cid:12)(cid:12) ≤ O (( R n ) H ; 2 , m , . . . , m n ) . Proof.
Take x + · · · + x n − r for the role of Q from Theorem 4.2 and pick r sothat v is generic with respect to ∂ B r . (cid:3) The Gottlieb and Gauss-Bonnet Equivariant Formulas
Our next goal is to reinterpret Gottlieb’s formulas ([G1]) for indices of pullbackvector fields within an equivariant setting.Let us recall the notion of a pullback field (see [G1]). Let F : X → Y be adifferentiable map of two n -dimensional Riemannian manifolds, and w a vectorfield on Y . Let F ∗ w be a field on X defined by the formula (cid:10) ( F ∗ w )( x ) , u ( x ) (cid:11) X = (cid:10) w, DF ( u ( x )) (cid:11) Y For example, take Q = a x d + · · · + a n x dn ∈ P Ψ (all a i >
0) plus a lower degree polynomial. , where x ∈ X , u ( x ) ∈ T x X is a generic vector, and (cid:10) ∼ , ∼ (cid:11) denotes the scalarproduct in the appropriate tangent space. In other words, if ω is a 1-form dual to w in Y , then F ∗ w is dual to F ∗ ω in X .In the case of a gradient field w = ∇ f on Y ( f : Y → R being a smooth function), F ∗ w = ∇ ( f ◦ F ).Note that when F is an equivariant map, both metrics on X and Y are G -invariant, and w is an invariant field, then F ∗ w is invariant as well.We recall the notion of an equivariant degree Deg G ( F ) ∈ A ( G ) of a G -map F : X → Y between two compact G -manifolds of the same dimension (cf. [L]).Crudely, it is an element of Q ( H ) ∈ conj ( G ) Z whose ( H )-component is the usual deg ( F H ), where F H : X H → Y H . In fact, such an element Deg G ( F ) ∈ A ( G ).This naive construction runs into some complications because of the ambiguities inchoosing orientations of fixed point components, both in the source and the target.In a sense, one wants to coordinate the orientations of X H and Y H (when they areof the same dimension). The issues with the coherent orientations can be resolvedby introducing some additional synchronizing structure called in [L] ”the O ( G )-transformation of the fiber transports”. Roughly speaking, it assigns a transfer G -map ( T F ( x ) Y ) c → ( T x X ) c to each x ∈ X .Fortunately, we need to employ Deg G ( F ) in a particular situation, where thesynchronization of the orientations can be achieved by pedestrian means. Consideran equivariant immersion f of a closed oriented ( n − G -manifold Z into an real n -dimensional space V of an orthogonal G -representation. Denoteby S ( V ) the unit sphere centered at the origin. To each x ∈ Z we assign the unitvector n ( x ) tangent to V at f ( x ) and normal to the f -image of a small neighborhood U x ⊂ Z of x . The orientations of V and Z help to resolve the two-fold ambiguityin picking n ( x ).Since f is an immersion, for any H ⊂ G , we get Z H = f − ( f ( Z ) ∩ V H ). More-over, because n ( x ) is orthogonal to f ( U x ) at f ( x ) ∈ V H , n ( x ) must be parallel to V H . Indeed, if n ( x ) would have a nontrivial component ν x normal to V H , ν x mustbe moved by elements of H \ { } ; on the other hand, n ( x ) is H -invariant since T f ( x ) ( U x ) is.In fact, f H : Z H → V H is an immersion as well. Therefore an orientation of V H , with the help of the ”field” n ( x ) (we assume that x ∈ Z H \ Sing ( f H )), picksa particular orientation of Z H . In the following, we assume that the orientationsof V H and Z H are always synchronized in this way.Thus the Gaussian map γ H : Z H → S ( V H ) is well-defined for any ( H ) that oc-curs as an orbit-type of Z . Unless dim ( V H ) ≤ S ( V H ) is connected. In such case,the degree deg ( γ H ) is defined as a sum of degrees of maps { γ Hα : Z Hα → S ( V H ) } α ,where Z Hα denotes a typical connected component of Z H . When dim ( V H ) = 1, S ( V H ) = a ` b , and deg ( γ H ) is defined as the sum of degrees of the two obviousmaps with the singleton targets a and b .The theorem below is an equivariant version of the ”Topological Gauss-BonnetTheorem” from [G, page 466]. Theorem 5.1.
Let V be a real vector space of dimension n on which a compact Liegroup G acts orthogonally. We assume that V admits an invariant non-vanishing vector field w . Let X a compact smooth n -dimensional G -manifold with an ori-ented boundary ∂ X . Consider a G -map F : X → V whose Jacobian is non-zeroon ∂ X . Let v = F ∗ w be the the pullback of w under F . Denote by { ∂ + k X } ≤ k ≤ n the Morse stratification of X induced by v .Then the degree of the Gauss map γ : ∂ X → S ( V ) with values in A ( G ) can becomputed by Deg G ( γ ) = χ G ( X ) − Ind G ( v ) = − n X k =1 ( − k χ G ( ∂ + k X )(5.1) Hence,
Ind G ( v ) and the RHS of (5 . are w -independent .Proof. Since DF | ∂ X is of the maximal rank, we can pullback the G -invariantriemannian metric g in V to an equivariant collar of ∂ X ⊂ X and then extendthe pullback F ∗ ( g ) to an invariant metric on X . Let n be the unitary field out-ward normal to ∂ X . Then, as we described prior to the statements of Theorem5.1, the Gauss map γ : x → DF ( n ( x )), x ∈ ∂ X , is well-defined, equivariant,and helps to pick coherent orientations of components in ∂ ( X H ). We have no-ticed already that n ( x ) is contained in T x ( X H ) and is normal to ∂ ( X H ), pro-vided x ∈ X H . Also, for x ∈ X H , v ( x ) ∈ T x ( X H ). We can apply the non-equivariant Gottlieb’s formula to each γ H : ∂ ( X H ) → S ( V H ) to conclude that deg ( γ H ) = χ ( X H ) − Ind ( v | X H ) = − P dim ( V H ) k =1 ( − k χ ( ∂ + k ( X H )). By Lemma 2.1,the latter sum is the ( H )-component ch H of − P nk =1 ( − k χ G ( ∂ + k ( X )) ∈ A ( G ).Finally, { deg ( γ H ) } ( H ) ∈ Φ( G ) detect Deg G ( γ ). Along the way, we have shown that Deg G ( γ ) ∈ A ( G ). (cid:3) Remark
Formula (5 .
1) tells us that we can define
Deg G ( γ ) ∈ A ( G ) as − n X k =1 ( − k χ G ( ∂ + k X )(5.2)for any choice of an invariant field w = 0 in V , thus a priori avoiding all the troubleswith the orientations. On the other hand, this definition of Deg G ( γ ) makes perfectsense for any equivariant immersion f : Z → V of a closed oriented G -manifold Z of codimension one: just define ∂ +0 Z to be the locus { x ∈ Z | (cid:10) n ( x ) , w ( x ) (cid:11) ≤ } ,and then proceed as in (1 . P n − k =0 ( − k χ G ( ∂ + k Z )is w -independent. To prove this conjecture will suffice to construct an equivariantcoboundary X for Z and to extend f : Z → V to a G -map F : X → V . Corollary 5.1.
Let g denotes the G -invariant Riemannian metric on V . Undernotations and hypotheses of Theorem 5.1 and we get Z ∂ X H K H dµ H = − dim ( V H ) X k =1 ( − k χ ( ∂ + k X H )(5.3) Here we are employing the pullback metric F ∗ ( g ) in vicinity of ∂ X ⊂ X to generatethe volume form dµ H on ∂ X H and its normal curvature K H . For example, such w = 0 exists when V = U ⊕ R , where U is the space of a G -representation:just put w = ∇ ( f ), where f : U ⊕ R → R is the obvious projection. Note that (5 .
1) proves that
Deg G ( γ ), a priori an element of Q ( H ) ∈ conj ( G ) Z , actuallybelongs to A ( G ). Probable Refinements of the Equivariant Morse Formula
One can refine the definition Birnside ring A ( G ) in a number of ways. Forexample, following Dieck [D1, I.10.3], one can introduce the component category Π ( G, X ) of a G -space X whose objects are G -maps x : G/H → X . A morphism σ from x : G/H → X to y : G/K → X is a G -map σ : G/H → G/K such that x and y ◦ σ are G -homotopic. Denote by Is Π ( G, X ) the set of isomorphism classes[ x ] of objects x : G/H → X . Let A G ( X ) := Z [ Is Π ( G, X )], where Z [ S ] stands fora free abelian group with the basis S . In fact, there is a bijection Is Π ( G, X ) → a ( H ) ∈ conj ( G ) W H \ π ( X H ) . This construction defines a covariant functor A G ( ∼ ) on the category of G -spaces X with values in the category of abelian groups, a functor which is sensitive to theconnected component structure of the sets { X H } . We conjecture that all previousresults can be restated in terms of A G ( ∼ ) along the lines of [LR].However, we would like to stress a different generalization of A ( G ) and to spec-ulate about the corresponding Morse Formulas. In this generalization one paysa close attention to the H -representations { ψ Hα } α arising in the normal bundles ν ( X H , X ).Let G be a compact Lie group. For each ( H ) ∈ Φ( G ), fix a set Ψ( H ) of dis-tinct isomorphism classes of representations { ψ Hα : H → GL ( V α ) } α with the prop-erty V Hα = { } . Moreover, for any ψ α ∈ Ψ( H ) and K ⊂ H , the representation˜ Res K ( ψ α ) : K → GL ( V α /V Kα ) is required to be in Ψ( K ). We call a collection F := { ψ α } ( H ) ∈ Φ( G ) , α ∈ Ψ( H ) of such representations a normal family.Let X Hψ a denote the set of points in X H with the normal representations isomor-phic to ψ α . Definition 6.1.
Let F = F ( G ) be a normal family of representations. Two com-pact smooth G -manifolds X and Y are F -equivalent if, for each ( H ) ∈ Φ( G ) , and ψ a ∈ Ψ( H ) ⊂ ( G ) , χ ( X Hψ a ) = χ ( Y Hψ a ) . We denote by A ( G, F ) the group of such equivalence classes. Conjecture 6.1.
All the equivariant Morse formulas above are valid in the refinedBurnside group A ( G, F ) . References [B] Bredon, G.,
Introduction to Compact Transformation Groups , Academic Press, New York -Lndon, 1972.[D] tom Dieck, T.,
The Burnside Ring of a Compact Lie Group. I , Math. Ann. 215, 235-250(1975).[G] Gottlieb, D.H.,
All the Way with Gauss-Bonnet and the Sociology of Mathematics , Math.Monthly, 103 (1996), 457-469.[G1] Gottlieb, D.H.,
On the Index of Pullback Vector Fields , Differential Topology Proceedings,Lecture Notes in Mathematics 1350, 167-170 (1988).[K] Katz, G.,
Convexity of Morse Stratifications and Spines of 3-Manifolds , preprint 2006,ArXiv: math.GT/0611005, v1.[K1] Katz, G.,
The Wall Groups of Finite Groups and the G-Signatures of Manifolds , Math.USSR Sbornik, 27: 2 (1975), 163-181. [Kh] Khovanskii, A. G.,
Index of a Polynomial Vector Field , Functional Analysis and its Appli-cations (Russ), 13:1 (1979), 49-58.[L] L¨uck, W.,
The Equivariant Degree , Proceedings of Algebraic Topology and TransformationGroups (G¨ottingen 1987), Lecture Notes in Mathematics 1361, 123-165, Springer-Verlag.[LR] L¨uck, W., Rosenberg, J.,
The equivariant Lefschetz fixed point theorem for proper cocompact G -manifolds [Mo] Morse, M. Singular points of vector fields under general boundary conditions , Amer. J.Math. 51 (1929), 165-178.[O] Robert Oliver, R.,
A Transfer for Compact Lie Group Actions
Proc. of AMS, Vol. 97, No.3 (Jul., 1986), pp. 546-548.
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