A note on the equivariant cobordism of generalized Dold manifolds
AA NOTE ON THE EQUIVARIANT COBORDISM OF GENERALIZEDDOLD MANIFOLDS
AVIJIT NATH AND PARAMESWARAN SANKARAN
Dedicated to Professor Daciberg Lima Gon¸calves on the occasion of his 70th birthday.
Abstract.
Let (
X, J ) be an almost complex manifold with a (smooth) involution σ : X → X such that Fix( σ ) (cid:54) = ∅ . Assume that σ is a complex conjugation, i.e, thedifferential of σ anti-commutes with J . The space P ( m, X ) := S m × X/ ∼ where ( v, x ) ∼ ( − v, σ ( x )) is known as a generalized Dold manifold. Suppose that a group G ∼ = Z s acts smoothly on X such that g ◦ σ = σ ◦ g for all g ∈ G . Using the action of thediagonal subgroup D = O (1) m +1 ⊂ O ( m + 1) on the sphere S m for which there are onlyfinitely many pairs of antipodal points that are stablized by D , we obtain an action of G = D × G on S m × X , which descends to a (smooth) action of G on P ( m, X ). Whenthe stationary point set X G for the G action on X is finite, the same also holds for the G action on P ( m, X ). The main result of this note is that the equivariant cobordismclass [ P ( m, X ) , G ] vanishes if and only if [ X, G ] vanishes. We illustrate this result in thecase when X is the complex flag manifold, σ is the natural complex conjugation and G ∼ = ( Z ) n is contained in the diagonal subgroup of U ( n ). Introduction
Recall that the classical Dold manifold P ( m, n ) is defined as the orbit space of the Z / Z -action on S m × C P n generated by the involution ( v, [ z ]) (cid:55)→ ( − v, [¯ z ]) , v ∈ S m , [ z ] ∈ C P n .Here [¯ z ] denotes [¯ z : · · · : ¯ z n ] when [ z ] = [ z : · · · : z n ] ∈ C P n . See [2].Let σ : X → X be a complex conjugation on an almost complex manifold ( X, J ), thatis, σ is an involution such that, for any x ∈ X, the differential T x σ : T x X → T σ ( x ) X satisfies the equation J σ ( x ) ◦ T x σ = − T σ ( x ) σ ◦ J x . See [1, § σ ) (cid:54) = ∅ .The generalized Dold manifold P ( m, X ) was introduced in [6] as the quotient of S m × X under the identification ( v, x ) ∼ ( − v, σ ( x )).We obtained a description of its tangent bundle, and, assuming that H ( X ; Z ) = 0, aformula for the Stiefel-Whitney classes of P ( m, X ). We obtained conditions for the (non)vanishing of the unoriented cobordism class [ P ( m, X )] ∈ N ∗ .Let G = ( Z ) n and consider the class of all smooth compact manifolds (without bound-ary) admitting (smooth) G -actions. Recall that two manifolds M , M with G -actions Mathematics Subject Classification.
Key words and phrases.
Generalized Dold manifolds, elementary abelian 2-group actions, equivariantcobordism, isotropy representations, flag manifolds.Most of this work was done when both the authors were at the Institute of Mathemtaical Sciences,Chennai. Both the authors were partially supported by a XII Plan Project, Department of AtomicEnergy, Government of India. a r X i v : . [ m a t h . A T ] F e b A. NATH AND P. SANKARAN are G - equivariantly cobordant if there exists a compact manifold-with-boundary W ad-mitting a smooth G action on it such that the boundary ∂W , with the restricted G -action, is equivariantly diffeomorphic to the disjoint union M (cid:116) M . The G -equivariantcobordism class of M with a given G -action φ is denoted [ M, φ ] or more briefly [
M, G ].In this note shall only consider G -actions on M with only finite stationary point set M G = { x ∈ M | g.x = x ∀ g ∈ G } . The set of all G -equivariant cobordim classes of com-pact G -mainfolds with finite stationary point sets is a graded Z -algebra, denoted Z ∗ ( G )in which addition corresponds to taking disjoint union and multiplication to the cartesianproducts (with the obvious G -actions). One has the forgetful map (cid:15) : Z ∗ ( G ) → N ∗ to thecobordism algebra sending [ M, φ ] to [ M ].We consider the action of a subgroup D of O (1) m +1 ⊂ O ( m + 1) on S m and assumethat there are at most finitely many pairs of antipodal points in S m that are stable by theaction of D . (This is evidently equivalent to the requitement that D act on R P m withonly finitely many stationary points.) For any G action on X that commutes with σ , weobtain an action of G := D × G on P ( m, X ) induced by the action of G on S m × X . If X G is finite, so is P ( m, X ) G .We now state the main result of this paper. Theorem 1.1.
Suppose that G ∼ = Z q acts on an almost complex manifold ( X, J ) such that X G is finite and that the isotropy representation at each G -fixed point is complex linear.Suppose that the action of D ⊂ ( O (1)) m +1 ⊂ O ( m + 1) on R P m has only finitely manystationary points. Then [ X, G ] = 0 ⇐⇒ [ P ( m, X ) , G ] = 0 where G = D × G . We illustrate, in §
4, the above result when X is a complex flag manifold.2. Equivariant cobordism and the representation ring
Let M d be a smooth compact manifold with a smooth G ∼ = ( Z ) q action with onlyfinitely many stationary points. Then [ M, φ ] ∈ Z ∗ ( G ) is determined completely by theisotropy representations of G at the stationary points of M . More precisely, denote by R ∗ ( G ) the representation ring of G whose elements are formal Z -linear combinations ofisomorphism classes of finite dimensional real representations of G . If U, V are two G -representations, the product [ U ] . [ V ] in R ( G ) is, by definition, the class of U ⊕ V (with thediagonal G -action). Note that R ( G ) is graded via the degree of the representation. Wemay identify R ∗ ( G ) with the polynomial algebra over Z with indeterminates v χ , χ ∈ ˆ G :=Hom( G, Z ), the group of characters of G . Under this isomorphism [ V ] corresponds tothe monomial v m χ · · · v m r χ r where V ∼ = R m χ ⊕ · · · ⊕ R m r χ r . Here R kχ denotes the direct sum of k copies of the 1-dimensional representation R χ ∼ = R on which G acts via the character χ .Given ( M, φ ), we obtain an element (cid:80) x ∈ M G [ T x M ] ∈ R ( G ), where T x M denotes tangentspace regarded as the isotropy representation of G . The map η ∗ : Z ∗ ( G ) → R ∗ ( G ) sending[ M, φ ] to (cid:80) x ∈ M G [ T x M ] is a well-defined algebra homomorphism. By a result of Stong [7], η ∗ is in fact a monomorphism. The finiteness of M G implies that T x M does not contain QUIVARIANT COBORDISM OF GENERALIZED DOLD MANIFOLDS 3 the trivial G - representation. So the image of η ∗ is contained in the subalgebra of R ∗ ( G )generated by the nontrivial one-dimensional representations of G .The diagonal subgroup D n ∼ = Z n of O ( n ) ⊂ U ( n ) acts the complex flag manifold U ( n ) /U ( n ) × · · · × U ( n r ) = C G ( n , . . . , n r ) with finitely many stationary points. Here n = (cid:80) ≤ j ≤ r n j . In fact, the stationary points are precisely the flags L = ( L , · · · , L r )where each component L j is spanned by a subset of the standard basis e , . . . , e n . Similarly, D m +1 ⊂ O ( m + 1) acts on S m . Although there is no point on the sphere which isstationary, the induced action on the real projective space R P m has m + 1 stationarypoints, namely, [ e ] , . . . , [ e m +1 ]. We obtain an action of D m + n +1 = D m +1 × D n on S m × C G ( n , . . . , n r ). This action yields an action of D m + n +1 on P ( m, C G ( n , . . . , n r )) withfinitely many stationary points, namely, [ e j , L ] , ≤ j ≤ m + 1 , with L as above. We shallconsider the restricted action of certain subgroups of D m + n +1 on P ( m, C G ( n , . . . , n r ))with finitely many stationary points and obtain results on the (non) vanishing of theequivariant cobordism classes of P ( m, C G ( n , . . . , n r )).3. Equivariant cobordism of generalized Dold manifolds.
Let (
X, J ) be an almost complex manifold and let σ be a complex conjugation withnon-empty fixed point set. We denote Fix( σ ) by X R . Suppose that G ∼ = Z q acts smoothlyon X such that (i) t ◦ σ = σ ◦ t for all t ∈ G , and, (ii) the stationary point set X G for the G action is finite. Then G acts on X R with X G R = X R ∩ X G .We have the following lemma which is a straightforward generalization of [1, Theorem24.4]. Lemma 3.1.
With the above notations, suppose that t ◦ σ = σ ◦ t for all t ∈ G and that,for each x ∈ X G R , the isotropy representations are C -linear; equivalently J x : T x X → T x X is G -equivariant. Then [ X, G ] = [ X R , G ] in Z ∗ ( G ) .Proof. Let x ∈ X G . Our hypothesis that t ◦ σ = σ ◦ t for all t ∈ G implies that T x σ : T x X → T σ ( x ) X is an isomorphism of G -modules. Indeed, for t ∈ G , we have T x σ ( t.v ) = T x σ ( T x t ( v )) = T σ ( x ) t ( T x σ ( v )) = t.T x σ ( v ) for all v ∈ T x X . In particular [ T x X ] + [ T σ ( x ) X ] =0 in R ∗ ( G ).Suppose that σ ( x ) = x , that is, x ∈ X R . We have T x X R ∩ J x ( T x X R ) = 0. To seethis, we need only observe that T x X R and J ( T x X R ) are the 1- and − T x σ : T x X → T x X . Since J x : T x X R → J x ( T x X R ) is G -equivariant, T x X R and J x ( T x X R )are isomorphic as G -modules and so [ T x X ] = [ T x X R ][ J x ( T x X R )] = [ T x X R ] in R ∗ ( G ).Now η ∗ ([ X, G ]) = (cid:80) x ∈ X G [ T x X ] = (cid:80) x ∈ X G R [ T x X ] + (cid:80) x ∈ X G \ X R [ T x X ] = (cid:80) x ∈ X G R [ T x X R ] ,since [ T σ ( x ) X ] cancels out [ T x X ] for each x ∈ X G \ X R . So η ∗ ([ X, G ]) = (cid:80) x ∈ X G R [ T x X R ] =( (cid:80) x ∈ X R [ T x X R ]) = η ∗ ([ X R ] ). Since η ∗ : Z ∗ ( G ) → R ∗ ( G ) is a monomorphism, we aredone. (cid:3) We remark that when J arises from a complex structure on X and G acts as a groupof biholomorphisms, J x is G -equivariant for all x ∈ X G . A. NATH AND P. SANKARAN
Suppose that for the restricted action of a subgroup D ⊂ D m +1 ⊂ O ( m + 1) on thesphere S m = O ( m +1) /O ( m ), the induced action on R P m has only finitely many stationarypoints. These points are precisely [ e j ] ∈ R P m , ≤ j ≤ m + 1.Consider the action of D × G on S m × X . We assume that the t ◦ σ = σ ◦ t for all t ∈ G so that the action of G := D × G on S m × X descends to an action on P ( m, X ).It is readily verified that P ( m, X ) G equals { [ e j , x ] | x ∈ X G R , ≤ j ≤ m + 1 } . Evidently,[ P ( m, X ) , G ] = 0 if [ P ( m, X ) , D m +1 × G ] = 0.Let x ∈ X G R . Since [ v, x ] = [ − v, σ ( x )] = [ − v, x ], we have a well-defined cross-section s x : R P m → P ( m, X ) defined by [ v ] (cid:55)→ [ v, x ]. In fact ([ v ] , x ) (cid:55)→ [ v, x ] is a well-definedimbedding s : R P m × X R → P ( m, X ) . We shall denote by ι j : X (cid:44) → P ( m, X ) the fibre-inclusion x (cid:55)→ [ e j , x ] , ≤ j ≤ m + 1. We note that, with the trivial G action on R P m understood, the embedding s x is G -equivariant. In fact, s is G - equivariant: if γ = ( α, t ) ∈ D × G , then, we see that s ( γ. ([ v ] , x )) = s ([ α.v ] , t.x ) = [ α.v, t.x ] = γ. [ v, x ] = γ.s ([ v ] , x ). Onthe other hand, ι j is not G -equivariant since e j is not D -fixed. However, it turns out that,after twisting the action of G on X , ι j becomes G -equivariant, as we shall now explain.Let χ (cid:48) j : D → (cid:104) σ (cid:105) be the homomorphism whose kernel equals the isotropy group D j ⊂ D at e j ∈ S m . Define the χ (cid:48) j -twisted action of G on X by ( α, t )( x ) := χ (cid:48) j ( α )( t.x ). We notethat both the G - actions agree on X R and that the G -action obtained from the restrictionof the action to p − ([ e j ]) of the twisted G -action on P ( m, X ) is the same as the that ofthe original G -action on X . We shall denote by X j ∼ = X the fibre p − ([ e j ]) ⊂ P ( m, X ).We now verify that ι j is G -equivariant with respect to the twisted G -action on X . Forthis purpose, let γ = ( α, t ) ∈ G , x ∈ X . Then ι j ( γ.x ) = ι j ( χ (cid:48) j ( α ) t.x ) = [ e j , χ (cid:48) j ( α ) t.x ]while γ. ( ι j ( x )) = γ. [ e j , x ] = [ αe j , t.x ]. If α ( e j ) = e j , then χ j ( α ) = 1 and so it followsthat ι j ( γ.x ) = γ. ( ι j ( x )). If α ( e j ) = − e j , then χ j ( α ) = σ and so γ ( ι j ( x )) = [ − e j , t.x ] =[ e j , σt.x ] = [ e j , χ (cid:48) j ( α ) t.x ] = ι j ( γ.x ), proving our claim. It follows that the ι j ∗ : T x X → T [ e j ,x ] P ( m, X ) is G -equivariant and so, we have a decomposition of the tangent space T [ e j ,x ] P ( m, X ) into G -submodules: T [ e j ,x ] P ( m, X ) = s x ∗ ( T [ e j ] R P m ) ⊕ ι j ∗ ( T x X ) = s x ∗ T [ e j ] R P m ⊕ T [ e j ,x ] X j . (1)Since x ∈ X R and since the twisted G -action on X R coincides with the untwisted action, itfollows that ι j ∗ T x X R ⊂ T [ e j ,x ] X j is isomorphic as a G -submodule to T x X R . We claim thatits G -complement ι j ∗ ( J x T x X R ) ⊂ T [ e j ,x ] X j is isomorphic as a G -module to E j ⊗ T x X R .Here E j = R e j the one dimensional D -representation corresponding to the character χ j .In fact, θ : E j ⊗ T x X R → ι j ∗ J x T x X R , defined by e j ⊗ u (cid:55)→ ι j ∗ ( J u ), is an isomorphism of G -modules. To see this, let γ = ( α, t ) ∈ D × G . We have γ.u = t.u = t ∗ ( u ) ∀ u ∈ T x X R .Also, γ.e j = χ j ( α ) e j = ± e j where the sign is positive precisely if α ∈ D j ⊂ D . Thus θ ( γ ( e j ⊗ u )) = θ ( ± e j ⊗ t ∗ u ) = ± ι j ∗ J ( t ∗ u ) = ± ι j ∗ t ∗ ( J u ) where the sign is positive preciselywhen α ∈ D j ⊂ D .On the other hand, since ι j ∗ is G -equivariant with respect to the twisted G -actionon T x X , γ ( θ ( e j ⊗ u )) = γ ( ι j ∗ J u ) = ι j ∗ ( γ.J u ) = ι j ∗ (( α, t ) . ( J u )) = ι j ∗ ( χ (cid:48) j ( α ) t ∗ ( J u )).Note that as u ∈ T x X R , J u is in the − σ ∗ . So, from the definition of QUIVARIANT COBORDISM OF GENERALIZED DOLD MANIFOLDS 5 χ (cid:48) j : D → (cid:104) σ (cid:105) , we have χ (cid:48) j ( α ) ∗ .J u = (cid:26) σ ∗ J u = − J u if α / ∈ D j J u if α ∈ D j . (2)Consequently, γθ ( e j ⊗ u ) = ± ι j ∗ ( t ∗ ( J u )) where the sign is positive precisely when α ∈ D j ⊂ D . Hence θ is a G -isomorphism. Therefore θ is G -isomorphism.We summarise the above discussion in the following. Proposition 3.2.
Suppose that t ∗ J x = J x t ∗ for all t ∈ G, x ∈ X G R . With the abovenotations, we have, for ≤ j ≤ m + 1 and x ∈ X G R , an isomorphism of G -modules: T [ e j ,x ] P ( m, X ) ∼ = T [ e j ] R P m ⊕ T x X R ⊕ ( E j ⊗ T x X R ) . (3) (cid:3) We apply the above proposition to identify the image of [ P ( m, X ) , G ] in R ( G ) under η ∗ .Since D is possibly a proper subgroup of D m +1 , the characters χ j are not necessarilylinearly independent. However the D -representations E j , ≤ j ≤ m + 1 , are pairwisenon-isomorphic; equivalently the characters χ j , ≤ j ≤ m + 1 , are pairwise distinct.Indeed, suppose that for some i (cid:54) = j , E i ∼ = E j , then D i = D j and so, for any a, b ∈ R , wehave χ. ( ae i + be j ) = ± ( ae i + be j ) ∀ χ ∈ D . This implies that, for any a, b not both zero,the point R ( ae i + be j ) ∈ R P m is D -fixed. This contradicts our hypothesis that R P m hasonly finitely many stationary points for the D -action. We shall write χ j to also denotethe isomorphism class [ E j ] ∈ R ( D ) of the irreducible representation of D . Notation.
Let k be a positive integer. We shall denote by [ k ] the set { , · · · , k } .Fix a basis t r , ≤ r ≤ q, for G . Then the character group ˆ G consists of elements y α , α ⊂ [ q ] , where y α ( t j ) = − j ∈ α . We abbreviate y { j } , y { i,j } to y j , y i,j , ≤ i, j ≤ q, i (cid:54) = j, respectively and denote y ∅ by y . The y α generate R ( G ) as a Z -algebra.An entirely analogous notation is used for R ( D ). The elements χ j , y r , ≤ j ≤ m + 1 , ≤ r ≤ q, form a basis for ˆ G and we have R ( G ) = R ( D ) ⊗ R ( G ).The representation ring has an additional structure arising from tensor product of rep-resentations. If A = E α , B = E β are 1-dimensional representations of R ( D ) correspondingto characters χ α , χ β , then A ⊗ B has character χ α ∆ β where α ∆ β stands for the symmetricdifference α ∪ β \ α ∩ β . Note that the group operation in ˆ D is given by symmetric differ-ence: χ α .χ β = χ α ∆ β . But we will avoid using the notation χ α .χ β ∈ ˆ D since the producthas been given a different meaning in R ( D ); instead we shall denote this character by χ α ⊗ χ β . In the more general case where [ A ] = χ α · · · χ α r , [ B ] = χ β · · · χ β s ∈ R ( D ), wehave [ A ⊗ B ] = (cid:81) ≤ i ≤ r, ≤ j ≤ s ( χ α i ⊗ χ β j ).We will need to consider tensor product representations of G of the form E ⊗ V where E, V are representations of D and G respectively with E being 1-dimensional. A. NATH AND P. SANKARAN
It will be convenient to identify χ α with χ α ⊗ y and y β with χ ⊗ y β for α ⊂ [ m +1] , β ⊂ [ q ]. Let [ E ] = χ ∈ R ( D ) and suppose that [ V ] = y β · · · y β d ∈ R ( G ). Then[ E ⊗ V ] = ( χ ⊗ y β ) · · · ( χ ⊗ y β d ) ∈ R ( G ).As is well-known, T [ e j ] R P m = E j ⊗ E ⊥ j ; here E ⊥ j = ⊕ ≤ i ≤ m +1 ,i (cid:54) = j E i ⊂ R m +1 , the orthog-onal complement to E j in R m +1 . (See [4].) This is in fact an isomorphism of G -moduleswhere the G action is via the projection G → D . So [ T [ e j ] R P m ] = (cid:81) ≤ i ≤ m +1 ,i (cid:54) = j χ i ⊗ χ j ∈ R ( G ) . The following proposition is now immediate from Proposition 3.2.We shall denote by f ( y β ) ∈ R ( G ) a polynomial in the variables y β , β ⊂ [ q ]. Proposition 3.3.
Suppose that G ∼ = Z q acts on ( X, J ) with finitely many stationarypoints such that that the isotropy representation is C -linear at each point of X G R . Supposethat the action of D ⊂ D m +1 on R P m has only finitely many stationary points. With theabove notation, set f p ( y β ) := [ T p X R ] ∈ R ( G ) , p ∈ X G R . Then η ([ P ( m, X ) , G ]) = (cid:88) p ∈ X G R (cid:88) ≤ j ≤ m +1 (cid:89) ≤ i ≤ m +1 ,i (cid:54) = j χ i ⊗ χ j .f p ( y β ) f p ( χ j ⊗ y β ) (4) where G = D × G . (cid:3) We now turn to the proof of Theorem 1.1.
Proof of Theorem 1.1.
It suffices to show that η ([ X, G ]) = 0 ⇐⇒ η ([ P ( m, X ) , G ]) = 0.First suppose that η ([ X, G ]) = 0. Then we have [ X R , G ] = 0 in view of Lemma 3.1and the fact that Z ∗ ( G ) ⊂ R ( G ) has no non-zero nilpotent elements. So the G -stationarypoints of X R occur in pairs p, q , p (cid:54) = q , such that T p X ∼ = T q X as G -modules. It followsthat E j ⊗ T p X ∼ = E j ⊗ T q X as G -modules. Hence T [ e j ,p ] P ( m, X ) ∼ = T [ e j ,q ] P ( m, X ) for every j ∈ [ m + 1] by Proposition 3.2. It follows that η ([ P ( m, X ) , G ]) vanishes.For the converse part, assume that [ P ( m, X ) , G ] = 0. This means that the thereis a fixed point free bijective correspondence ( i, p ) ↔ ( j, q ) on P ( m, X ) G such that T [ e i ,p ] P ( m, X ) ∼ = T [ e j ,q ] P ( m, X ) as G -modules. That is, T [ e i ] R P m ⊕ T p X R ⊕ E i ⊗ T p X R ∼ = T [ e j ] R P m ⊕ T q X R ⊕ E j ⊗ T q X R (5)as G = D × G -module, by Proposition 3.2.Restricting to D = D × ⊂ G , we obtain T [ e i ] R P m ⊕ R n ⊕ E i ⊗ R n = T [ e j ] R P m ⊕ R n ⊕ E j ⊗ R n , (6)where R n is the trivial D -representation of degree n .Since T [ e i ] R P m ⊕ R = ( ⊕ i (cid:54) = k E k ⊗ E i ) ⊕ E i ⊗ E i = V ⊗ E i where V = ⊕ ≤ k ≤ m +1 E k ,cancelling R n − on both sides and using E i ⊗ E i = R we obtain from (6) that ( V ⊗ E i ) ⊕ ( E i ⊗ R n ) = ( V ⊗ E j ) ⊕ ( E j ⊗ R n ). That is,( V ⊕ R n ) ⊗ E i = ( V ⊕ R n ) ⊗ E j (7) QUIVARIANT COBORDISM OF GENERALIZED DOLD MANIFOLDS 7 as D -modules. Let k = dim R Hom D ( R , V ) = dim R Hom D ( E i , V ⊗ E i ), the multiplic-ity of the trivial (1-dimensional) submodule in V and let l = dim R Hom( E i ⊗ E j , V ) =dim R Hom D ( E i , V ⊗ E j ). As has been observed already, multiplicity of any one-dimensionalrepresentation occurring in V is at most 1 and so, in particular, k, l ≤ i (cid:54) = j , so that E i , E j ⊂ V are not isomorphic. Comparing the multiplicitiesof of E i on both sides of the above isomorphism we get k + n = l . This is a contradictionsince n ≥ l ≤ . So we must have i = j .Since ( i, p ) (cid:54) = ( j, q ) = ( i, q ) , we have p (cid:54) = q . Thus, fixing i = 1, we obtain a fixedpoint free bijection p ↔ q of X G R where q is such that (1 , p ) ↔ (1 , q ). Now restrictingthe G -isomorphism to G , we see that T p X R ∼ = T q X R . It follows that [ X R , G ] = 0 and so[ X, G ] = 0 by Lemma 3.1. (cid:3)
Remark 3.4. (i) Suppose that there exists an involution θ : P ( m, X ) → P ( m, X ) whichcommutes with each γ ∈ G . Let (cid:101) G be the group G ∪ { θ ◦ γ | γ ∈ G} . If P ( m, X ) (cid:101) G = ∅ ,then [ P ( m, X ) , (cid:101) G ] = 0 in Z ∗ ( (cid:101) G ) and so [ P ( m, X ) , G ] = 0 in Z ∗ ( G ).(ii) More generally, if H ∼ = Z p acts smoothly on P ( m, X ) such that θ ◦ γ = γ ◦ θ for all θ ∈ H, γ ∈ G . Then (cid:101) G = H × G acts on P ( m, X ). If P ( m, X ) (cid:101) G = ∅ , then [ P ( m, X ) , G ] = 0in Z ∗ ( G ) since [ P ( m, X ); (cid:101) G ] = 0.(iii) Let V be an ( m + 1)-dimensional D -representation where the multiplicity of eachone-dimensional representation occurring in V equals 1 so that the induced D -action on R P m has only finitely many stationary points. Using the observation that T [ e i ] R P m ⊕ R ∼ = V ⊗ E i as in the course of the above proof we see that, given distinct positiveintegers i, j ≤ m + 1 the D -representations T [ e i ] R P m , T [ e j ] R P m are isomorphic if and onlyif V ⊗ E j ∼ = V ⊗ E i , if and only if, for each k ≤ m + 1 , there exists an l ≤ m + 1 suchthat E k ⊗ E i ∼ = E l ⊗ E j ; equivalently χ k ⊗ χ i = χ l ⊗ χ j . Therefore, [ R P m , D ] (cid:54) = 0 ifand only if, for some i, k ≤ m + 1 , i (cid:54) = k, we have χ k ⊗ χ i (cid:54) = χ l ⊗ χ j for all l and any j ≤ m + 1 , j / ∈ { i, k } .4. Group actions on P ( m ; n , . . . , n r ) with finite stationary point sets Let G = Z q with standard Z -basis t j , ≤ j ≤ q and y j , ≤ j ≤ q, denote the basis forthe dual ˆ G = Hom( G, C × ) ∼ = Z q . We shall use multiplicative notation for group operationin G and ˆ G . The elements of G and ˆ G are labeled by subsets of [ q ] := { , . . . , q } where t α = (cid:81) j ∈ α t j , y α ( t j ) = − j ∈ α , α ⊂ [ q ]. It follows that y α ( t β ) = − α ∩ β ) is odd.Let V R = R G denote the regular representation of G and V = V R ⊗ C its complexifica-tion. Then V ∼ = C q decomposes into 1-dimensional complex representations as follows: V = ⊕ y ∈ ˆ G C y ; the G -action on C y is via the character y , that is, t.z = y ( t ) z ∀ t ∈ G .Let n ≤ q and let n , . . . , n r be positive integers such that n = (cid:80) ≤ i ≤ r n i . Let U ⊂ V be an n -dimensional complex G -submodule. The G -action on U ∼ = C n yields an action φ on the complex flag manifold X := C G ( n , . . . , n r ) ∼ = U ( n ) / ( U ( n ) × · · · × U ( n r )) which A. NATH AND P. SANKARAN is identified with the space of flags L := ( L , · · · , L r ) where the L j are C -vector subspacesof U such that dim L j = n j , ≤ j ≤ r, and L i ⊥ L j if i (cid:54) = j .The tangent bundle of X has the following description as a complex vector bundle: Let ξ j denote the complex vector bundle over X whose fibre over a flag L ∈ X is the vectorspace L j , ≤ j ≤ r . We shall denote by ¯ ξ j complex conjugate of ξ j . The fibre of ¯ ξ j over L is the vector space ¯ L j ⊂ C n . Then, by [3], τ X = (cid:77) ≤ i 2. Let S ⊂ ˆ G be any subset which does not contain the trivialrepresentation and S = n . It was shown in [5, Example 2.3] that if k < n odd, QUIVARIANT COBORDISM OF GENERALIZED DOLD MANIFOLDS 9 then [ R G ( k, n − k ) , φ S ] = 0. We consider here the more general case of a flag manifold X R = R G ( n , . . . , n r ) , r ≥ , where we assume that n is odd. We claim that, as in thecase of the Grassmannian, [ X R , φ S ] = 0. Although this is a routine generalisation of thecase of Grassmann manifolds, we give most of the details. Proof of Claim: In view of our hypotheses, S leaves out exactly one nonempty subset [ q ]which we shall denote by γ . As in [5], we shall denote the symmetric difference of two sets α, β by α + β and exploit the Boolean algebra structure on the power set of [ q ]. Note that α (cid:54) = α + γ for any α since γ (cid:54) = ∅ . Also α (cid:55)→ α + γ is an involutive bijection S → S since ∅ , γ are not in S . If A ⊂ S then we shall denote by A γ its image { α + γ | α ∈ A } ⊂ S underthis bijection. We see that, if A is odd, then A (cid:54) = A γ . If E ⊂ U R is spanned by standardbasis vectors e α , . . . e α k , we shall denote by E γ the span of e γ + α j , ≤ j ≤ k . Note that E γ (cid:54) = E if k = dim E is odd. Suppose that E = ( E , · · · , E r ) ∈ X R is a stationary pointof φ R S . Then so is E γ := ( E γ , . . . , E γr ). Moreover, since n = dim E is odd, we see that E γ (cid:54) = E . So E (cid:55)→ E γ is a fixed point free involution on the set of stationary points of X R .We shall show that T E X R ∼ = T E γ X R as G -representations. The claim follows from thissince η ∗ : Z ∗ ( G ) → R ( G ) is a monomorphism. As in the case of complex flag manifolds,the tangent bundle τ X R has the following description, due to Lam [3]. τ X R ∼ = (cid:77) ≤ i Differentiable periodic maps. Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag, Berlin, 1964.[2] Dold, Albrecht Erzeugende der Thomschen Algebra N . Math. Zeit. (1956) 25–35.[3] Lam, Kee Yuen A formula for the tangent bundle of flag manifolds and related manifolds. Trans.Amer. Math. Soc. (1975), 305–314.[4] Milnor, J. W.; Stasheff, J. D. Characteristic classes. Annals of Mathematics Studies, , PrincetonUniversity Press, Princeton, N. J. 1974.[5] Mukherjee, Goutam; Sankaran, Parameswaran Elementary abelian 2-group actions on flag manifoldsand applications. Proc. Amer. Math. Soc. (2) (1998) 595–606.[6] Nath, Avijit; Sankaran, Parameswaran On generalized Dold manifolds. Osaka J. Math. (2019),75–90. Errata , (2020) to appear.[7] Stong, R. E. Equivariant bordism and ( Z ) k actions. Duke Math. J. (1970) 779–785. Indian Institute of Science Education and Research, Tirupati, Rami Reddy Nagar,Karakambadi Road, Mangalam (P.O.), Tirupati 517507 E-mail address : [email protected] Chennai Mathematical Institute, H1 SIPCOT IT Park, Siruseri, Kelambakkam 603103 E-mail address ::