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Ω ∗ G ( X ) G or- by G is In the early 1960s, Conner and Floyd ([14, 15, 16]) began the study of geometric equivariant bordism theory by combining the ideas of bordism theory and transfor- mation groups. They studied geometric equivariant unoriented, oriented and unitary bordism theories for smooth closed manifolds with periodic diffeomorphisms, and the subject has also continued to further develop by extending their ideas or combining with various ideas and theories from other research areas since then. For example, tom Dieck [38] introduced and studied the homotopy theoretic equivariant bordisms GG G∗ M ∗ (X) and M G(X) for a G-space X where G is a compact Lie group, and recently, Buchstaber–Panov–Ray in [10] introduced and studied the universal toric G,G genus in a geometric manner. The geometric equivariant bordism ring Ω∗ (X) for a G- space X can also be defined in a natural way, and it is formed by singular G-manifolds f : M −→ X where each M is a smooth closed G-manifold with G-structure such that the G-action preserves the G-structure (in this case, we say that M admits a G- G,G G,G equivariant G-structure). When X is a point, Ω∗ (X) will be denoted by Ω∗ , and G it is an equivariant analogue of Ω∗ . It should be pointed out that in the above defini- tion, the G-equivariant G-structure of a smooth closed G-manifold with G-structure is equipped on the stably G-equivariant tangent bundle of M rather than the stably G- equivariant normal bundle of the embedding of M in some G-representation because there is a substantial difference between both notions (see [21]). In this paper we will pay more attention to the case in which G = U (i.e., G-unitary bordism). A nice survey of the development of G-unitary bordism is contained in the introduction of the paper [10] by Buchstaber–Panov–Ray.

Comparing with the nonequivariant case, a natural question arises and is stated as follows:

Question 1.1. What kinds of equivariant characteristic numbers determine the equivariant geometric bordism class of a smooth closed G-manifold with G-structure?

As far as the authors know, the above question is far from solved. Most known works with respect to the above question have considered mainly the cases (G,G) = k k (O, (Z2) ), (U, T × Zm), (SO, finite group) (e.g., see [7, 20, 22, 23, 27, 30, 36, 39, 40]). In the following, we mainly give an investigation in the cases G = O, U.

tom Dieck first investigated the above question. In his series of papers ([38, 39, 40]), by combining the geometric approach of Conner–Floyd ([15, 16]) and the K-theory ap- proach developed by Atiyah, Bott, Segal and Singer ([4, 5, 6]), tom Dieck introduced the bundling transformation which further develops ideas of Boardman [8] and Con- ner [13], and proved a series of integrality theorems in equivariant (co)bordism theory. k Furthermore, he gave an answer to the above question in the cases (G,G)=(O, (Z2) ), k (U, T × Zm), which is stated as follows:

k k O,(Z2) Theorem 1.2 (tom Dieck). When (G,G)=(O, (Z2) ), β ∈ Ω∗ is zero if and only if all k U,G equivariant Stiefel–Whitney numbers of β vanish. When (G,G)=(U, T × Zm), β ∈ Ω∗ is zero if and only if all equivariant K-theoretic Chern numbers of β vanish. 2 In their book [20, Appendix H], Guillemin–Ginzburg–Karshon discussed the prob- G lem of calculating the ring H∗ of equivariant Hamiltonian bordism classes of all uni- 1 tary Hamiltonian G-manifolds with integral equivariant cohomology classes 2π [ω −Φ], G where G is a torus. With respect to the determination of the ring H∗ , they posed three series of questions, the first one of which is stated as follows: Question 1.3 ([20, Question (A), §1.1, Appendix H]). Do mixed equivariant characteristic numbers form a full system of invariants of Hamiltonian bordism? On this question, Guillemin–Ginzburg–Karshon constructed a monomorphism G U,G ΣG : H∗ −→ Ω∗+2, so that Question 1.3 is equivalent to asking if the integral equivariant cohomology Chern numbers determine the equivariant geometric unitary bordism classes for the U,G ring Ω∗ . They showed that a closed unitary G-manifold M with only isolated fixed- U,G points represents the zero element in Ω∗ if and only if all integral equivariant coho- mology Chern numbers of M vanish, which gives a partial solution of Question 1.3. Furthermore, they posed the following conjecture without the restriction of isolated fixed-points. U,G Conjecture 1.4 ([20, Remark H.5, §3, Appendix H]). Let G be a torus. Then β ∈ Ω∗ is zero if and only if all integral equivariant cohomology Chern numbers of β vanish. For the detecting problem, it is natural to analyze the topological data of fixed point sets. This approach has been used successfully in the case when the fixed point sets are isolated ([20]). For the general case, we found that the analysis of fixed data seems to be not easy to handle and we turned to consider the geometric presentation of the image Φ([M]) ∈ MU∗(BG) of the universal toric genus instead, which admits a natural geometric representation described by Buchstaber–Panov–Ray in [10]. In particular, by using the Kronecker pairing, our argument can directly be reduced to the simpler calculation, so that the Boardman map as used in the tom Dieck’s work ([39, 40]) will not be involved yet. 1.2. Main results. The motivation of this paper is mainly supplied by the work of Guillemin–Ginzburg–Karshon as mentioned above. The geometric description of the universal toric genus developed by Buchstaber–Panov–Ray in [10] provides us much more insight, so that we can carry out our work by utilizing the Kronecker pairing of bordism and cobordism. This leads us to give an affirmative answer to Conjecture 1.4, which is stated as follows. Theorem 1.5. Conjecture 1.4 holds. As a further consequence, we obtain a satisfactory solution of Question 1.3. Corollary 1.6. Mixed equivariant characteristic numbers determine equivariant Hamiltonian bordism. It is interesting that our approach above can also be applied to the detecting prob- k lem of (Z2) -equivariant unoriented bordism, and in particular, it can still derive the k classical result of tom Dieck in the case (G,G)=(O, (Z2) ) of Theorem 1.2 by replacing the Boardman map by Kronecker pairing in the original proof of tom Dieck [39]. 3 This paper is organized as follows. In Section 2, we shall review Quillen’s geometric interpretation of homotopic unitary cobordism theory first, which is of essential impor- tance, so that the Kronecker pairing between bordism and cobordism can be calculated in a geometric way. We then review the universal toric genus defined by Buchstaber– Panov–Ray, which can be expressed in terms of Quillen’s geometric interpretation. We also recollect some necessary facts about equivariant Chern classes and equivariant Chern numbers. Then we shall give the proof of Theorem 1.5 in Section 3, which is more geometric. In particular, our approach in the unitary case can also be carried out k in the study of (Z2) -equivariant unoriented bordism, as we shall see in the final part of Section 3. Acknowledgements. The authors would like to thank Peter Landweber for his com- ments and suggestions on an earlier version of this paper. We are also grateful to the referee for carefully reading this manuscript and providing a number of valuable thoughts and helpful suggestions, which led to this version.

2. PRELIMINARIES 2.1. Geometric interpretation of elements in MU ∗(X) and Kronecker pairing. Given ∗ a topological space X, let MU∗(X) and MU (X) be the complex (homotopic) bordism and cobordism of X, which are defined as 2l+∗ MU∗(X) = lim [S ,X+ ∧ MU(l)] l−→∞ and ∗ 2l−∗ MU (X) = lim [S ∧ X+, MU(l)] l−→∞ respectively, where X+ denotes the union of X and a disjoint point, and MU(l) denotes the of the universal complex l-dimensional vector bundle over BU(l).

Geometrically, it is very well-known that elements of MUn(X) are regarded as bor- dism classes of maps M −→ X of stably complex closed n-dimensional manifolds M ∼ U to X since MUn(X) = Ωn (X). On the geometric interpretation of elements in MU ∗(X), Quillen showed in [33, Proposition 1.2] that for a manifold X, MU ±n(X) is isomorphic to the group formed by cobordism classes of proper complex-oriented maps f : M −→ X of dimension ∓n, where dim X − dim M = ±n. Recall that when n is even, a complex-oriented map f : M −→ X of dimension ∓n is a composition od maps of manifolds i π (2.1) M −−−→ E −−−→ X where π : E −→ X is a complex vector bundle over X, and i : M −→ E is an embedding endowed with a complex structure on its normal bundle. When n is odd, the complex orientation of f will be defined in a same way as above with E replaced by E × R in (2.1). It is very well-known that as generalized cohomology and homology theory, MU ∗(X) and MU∗(X) admit a canonical Kronecker pairing n h , i : MU (X) ⊗ MUm(X) −→ MUm−n ∼ U where MUm−n = MUm−n(pt) = Ωm−n. 4 This Kronecker pairing can be calculated in a geometric way as follows. For exam- ple, if α ∈ MU −n(X) is represented by a smooth fiber bundle of closed smooth stably complex manifolds E −→ X with dim E − dim X = n and β ∈ MUm(X) can be repre- U sented by a smooth map f : M −→ X, then the Kronecker pairing hα, βi ∈ Ωm−n is the ∗ bordism class of the pull-back f (E)= E ×X M as shown in the following diagram e fe E ×X M −−−→ E     y f y M −−−→ X (cf. [11, D.3.4, Appendix D]).

2.2. The universal toric genus. Based upon the works of tom Dieck, Krichever and Löffler (see [38, 40, 25, 28]), Buchstaber–Panov–Ray in [10] defined the universal toric genus Φ in a geometric manner, which is a ring homomorphism from the geometric unitary T k-bordism ring to the complex cobordism ring of the BT k

U,T k ∗ k Φ : Ω∗ −→ MU (BT ).

U ∗ k U Note that, as an Ω∗ -algebra, MU (BT ) is isomorphic to Ω∗ [[u1, ··· ,uk]] where ui is MU the cobordism c1 (ξi) of the conjugate Hopf bundle ξi over the i-th factor k ∗ k U of BT for i =1, ..., k, so MU (BT ) can be replaced by Ω∗ [[u1, ··· ,uk]]. U,T k k Let [M]T k ∈ Ωn be an element represented by a closed unitary T -manifold M. Buchstaber–Panov–Ray showed that Φ([M]T k ) can be defined to be the cobordism class k k k of the complex oriented map π : ET ×T k M −→ BT . More precisely, choose a T - equivariant embedding i : M ֒→ V into a unitary T k-representation space V . Then the Borelification of i gives a complex-oriented map

k k k (πl : ET (l) ×T k M ֒→ ET (l) ×T k V −→ BT (l

−n k k l k which determines a cobordism class αl in MU (BT (l)), where BT (l)=(CP ) and k 2l+1 k k k k k ET (l) =(S ) . Since BT = ∪lBT (l) and ET = ∪lET (l), these cobordism α α = lim α MU −n(BT k) classes l form an inverse system. This produces a class ←− l in , which is represented geometrically by the complex-oriented map

k k k . π : ET ×T k M ֒→ ET ×T k V −→ BT

Then Φ([M]T k ) is defined as this limit α. The following result is essentially due to Löffler [28] and Comezana [12] as noted by Hanke in [21]. e

U,T k ∗ k Proposition 2.1. The ring homomorphism Φ : Ω∗ −→ MU (BT ) is injective.

k U,T k Corollary 2.2. If M is a closed unitary T -manifold, then M is null-bordant in Ω∗ if and k k only if the complex-oriented map π : ET ×T k M −→ BT represents the zero element in MU ∗(BT k). 5 2.3. Equivariant Chern classes and equivariant Chern numbers. Let M be a closed unitary T k-manifold. Then applying the Borel construction to the stable tangent bundle k k T M of M gives a vector bundle ET ×T k T M over ET ×T k M. Definition 2.3. The total equivariant cohomology Chern class of M is defined to be the k k total Chern class of the vector bundle ET ×T k T M over ET ×T k M, i.e.

T k k c (M): = c(ET ×T k T M) k k = 1+ c1(ET ×T k T M)+ c2(ET ×T k T M)+ ··· T k T k ∗ k ∗ = 1+ c1 (M)+ c2 (M)+ ···∈ H (ET ×T k M)= HT k (M). k ∗ ∗ T k k k Let p : M −→ pt be the constant map, and let p! : H (ET ×T M) −→ H (BT ) be the equivariant Gysin map induced by p. Then, the integral equivariant cohomology Chern numbers of M are defined to be:

T k T k T k cω [M]T k := p! (cω (M)), each of which is a homogeneous polynomial of degree 2|ω| − dim M in the polynomial ∗ k ring H (BT ) = Z[x1, ..., xk] with deg xi = 2, where ω = (i1, i2, ··· , is) is a partition of T k T k T k T k |ω| = i1 + i2 + ··· + is, and cω (M) means the product ci1 (M)ci2 (M) ··· cis (M). k k Note that the map π : ET ×T k M −→ BT is the Borelification of p : M −→ pt, so T k p! is often replaced by the Gysin map π! in this paper.

3. PROOFOFMAINTHEOREMANDEQUIVARIANTUNORIENTEDBORDISM 3.1. Determine equivariant cohomology Chern numbers by ordinary Chern num- k ∗ T k k Z bers. Consider the equivariant cohomology Chern numbers cω [M]T ∈ H (BT ; ) ∗ k ∼ for the partition ω =(i1, i2, ··· , is) of weight |ω| = i1 + i2 + ··· + is. Since H (BT ; Z) = Z T k [x1, x2, ··· , xk] with deg xi = 2, the equivariant Chern number cω [M]T k admits the form: T k ω J cω [M]T k = X nJ x , J where the k-tuples of natural numbers J = (j1, j2, ..., jk) ranges over all k-partitions J j1 j2 jk ω Z of weight |ω| − n and x means x1 x2 ··· xk , and the coefficient nJ ∈ . It follows T k that to determine the equivariant cohomology Chern numbers cω [M]T k is equivalent ω to determine the coefficient nJ for each J =(j1, j2, ..., jk). ω The coefficient nJ is closely related to the ordinary Chern numbers of certain man- k i 2j +1 k ifolds. Indeed, for J = (j1, j2, ..., jk), let (Qi=1 S ) ×T M be the pull-back of the k C ji k : natural inclusion Qi=1 P ֒→ BT f k i fJ 2j +1 k k k (Qi=1 S ) ×T M −−−→ ET ×T M πJ  π   y fJ y k C ji k Qi=1 P −−−→ BT First, we find that: 6 Lemma 3.1. k ω J 2ji+1 nJ x =(πJ )!(cω((Y S ) ×T k T M)), i=1 where (πJ )! is the Gysin map induced by πJ . Proof. By the definition of equivariant Chern numbers, T k k ω J ∗ k Z cω [M]T k = π!(cω(ET ×T k T M)) = X nJ x ∈ H (BT ; ). J ∗ k i 2j +1 k ˜ k k Since (Qi=1 S ) ×T T M = fJ (ET ×T T M) and Gysin maps are commutative in ∗ ˜ ∗ the pull-back square, i.e, fJ π! =(πJ )!fJ , then we have: ω J ∗ k n x = f π!(cω(ET ×T k T M)) J J ∗ ˜ k = (πJ )!fJ cω(ET ×T k T M) k i 2j +1 k = (πJ )!(cω((Qi=1 S ) ×T T M)).  ′ ′ ′ ′ ′ Let ω = (i1, i2, ··· , is) and ω = (i1, i2, ··· , is) be two s-partitions, we say ω 6 ω ′ ′ ′ ′ provided ij 6 ij for each j and we say ω < ω if ω 6 ω and ω =6 ω. We also denote ′ ′ ′ ′ ω ± ω the partition (i1 ± i1, i2 ± i2, ··· , is ± is). ω For each partition ω and each k-partition J, the corresponding coefficient nJ is deter- ′ k i 2j +1 k ω mined by the ordinary Chern number cω[(Qi=1 S ) ×T M] and the coefficients nJ′ with J ′ < J,ω′ <ω. More precisely, we have: Proposition 3.2. When dimM > 0, one has: k ′ ′′ 2ji+1 ω ω ω cω[(Y S ) ×T k M]= nJ + X nJ′ m(J−J′), i=1 ω′+ω′′=ω ω′<ω,ω′′<ω,J′

J−J′ where mJ−J′ is the coefficient of x in the Chern class: k k ji ∗ ji j1+1 jk+1 ′′ C C Z Z cω (Y P ) ∈ H (Y P ; )= [x1, x2, ··· , xk]/(x1 , ··· , xk ). i=1 i=1

k i 2j +1 k Proof. The stable tangent bundle of (Qi=1 S ) ×T M admits a decomposition: k k k 2ji+1 2ji+1 ∗ ji k k C T ((Y S ) ×T M)=((Y S ) ×T T M) ⊕ πJ T (Y P ). i=1 i=1 i=1 By Cartan formula, we have: k k k 2ji+1 2ji+1 ∗ ji k k C cω((Y S ) ×T M)= cω((Y S ) ×T T M)+ πJ cω(Y P ) i=1 i=1 i=1

k k 2ji+1 ∗ ji ′ k ′′ C + X cω ((Y S ) ×T T M)πJ cω (Y P ). ω′+ω′′=ω i=1 i=1 ω′<ω,ω′′<ω 7 Taking (πJ )! to both sides, and we obtain:

k i 2j +1 k J π cω[(Qi=1 S ) ×T M]x ( J )! is isomorphic on the top cohomology k i 2j +1 k = (πJ )!cω((Qi=1 S ) ×T M) k i ∗ k i 2j +1 k C j = (πJ )!(cω((Qi=1 S ) ×T T M))+(πJ )!(πJ cω(Qi=1 P )) k ji ∗ k ji ′ ′′ ′ 2 +1 k ′′ C + ω +ω =ω (πJ )!(cω (( i=1 S ) ×T T M)πJ cω ( i=1 P )) Pω′<ω,ω′′<ω Q Q ω J k C ji = nJ x +(πJ )!(1)cω(Qi=1 P ) (πJ )!(1)=0, when dimM>0 k i k i ′ ′′ ′′ C j ′ 2j +1 k + ω +ω =ω cω ( i=1 P )(πJ )!(cω (( i=1 S ) ×T T M)). Pω′<ω,ω′′<ω,J′

k ′ ′′ 2ji+1 ω ω ω cω[(Y S ) ×T k M]= nJ + X nJ′ m(J−J′). i=1 ω′+ω′′=ω ω′<ω,ω′′<ω,J′

Clearly, one has

′ k 2ji+1 ω ′ ′ ′ ′ k ′ Z 6 6 Corollary 3.3. cω[(Qi=1 S ) ×T M] ∈ (nJ ) ⊂ with ω ω,J = (j1, ··· , jk) J, ω′ Z ω′ where (nJ′ ) is the ideal in generated by the integers nJ′ . ω On the contrary, when dimM = n > 0, the coefficient nJ of the equivariant Chern k k ′ T k ′ 2ji+1 k number cω [M]T is determined by the ordinary Chern numbers cω [(Qi=1 S ) ×T k 2j′+1 ′ ′ i k 6 6 M] of the manifolds (Qi=1 S ) ×T M with partition ω ω and J J. Indeed, we have:

ω k 2j˜i+1 k Z 6 ˜ 6 Proposition 3.4. nJ ∈ (cω˜ [(Qi=1 S ) ×T M]) ⊂ with ω˜ ω, J J, where the ideal k ˜i k ˜i 2j +1 k Z 2j +1 k (cω˜ [(Qi=1 S ) ×T M]) in is generated by the integers cω˜ [(Qi=1 S ) ×T M]. ω Proof. Note that when the weight |ω| < n, nJ = 0 for all J. When the weight |ω| = n, k i ω 2j +1 k the corresponding k-partition is J0 = (0, ··· , 0) and nJ0 = cω[(Qi=1 S ) ×T M]. When |ω| > n, by Proposition 3.2, we see

k ′ ω 2ji+1 ω k Z nJ − cω[(Y S ) ×T M] ∈ (nJ′ ) ⊂ i=1 8 with ω′ < ω,J ′

k ω′ j′′ ′′ 2 i +1 Z (nJ′ ) ⊂ (cω [(Y S ) ×T k M]) ⊂ , i=1 with ω′′ 6 ω′,J ′′ 6 J ′. Hence, we have: k k ′ ω 2ji+1 ω 2j˜i+1 k k Z nJ ∈ cω[(Y S ) ×T M]+(nJ′ ) ⊂ (cω˜ [(Y S ) ×T M]) ⊂ i=1 i=1 with ω˜ 6 ω, J˜ 6 J. 

T k Theorem 3.5. The equivariant Chern number cω [M]T k =0 for all the partition ω if and only k i 2j +1 k if the ordinary Chern number cω[(Qi=1 S )×T M]=0 for all ω and all J =(j1, j2, ..., jk). Proof. When dimM = 0, M is the disjoint union of some isolated points with triv- ial torus action, and the statement certainly holds. Now we assume dimM > 0. k T k ω˜ ˜ Suppose cω [M]T = 0 for all ω, then nJ˜ = 0 for all ω˜ and J. By Proposition 3.4, k 2ji+1 ω˜ c [( S ) × k M] ∈ (n ) = (0) ⊂ Z  ω Qi=1 T J˜ . The inverse direction is similar. 3.2. Proof of Theorem 1.5. Let M be a closed unitary T k-manifold. Since the universal toric genus U,T k ∗ k Φ : Ω∗ −→ MU (BT )

is injective, it is sufficient to show that the image Φ([M]T k )=0 if and only if all integral equivariant cohomology Chern numbers of M vanish. ∗ k Geometrically, the image Φ([M]T k ) ∈ MU (BT ) is represented by the map π : k k ET ×T k M −→ BT . Since the Kronecker pairing ∗ k k MU (BT ) ⊗ MU∗(BT ) −→ MU∗. defines an isomorphism ∗ k ∼ k MU (BT ) = HomMU∗ (MU(BT ), MU∗), ∗ k because MU (BT ) is a free MU∗-module in even dimensional generators. Therefore, the image Φ([M]T k ) is zero if and only if its evaluation with all the MU∗ k generators of MU∗(BT ) is zero. n 1 According to [1, 24], the natural inclusion maps CP ֒→ BT are the MU∗-generators 1 k C ji k of MU∗(BT ), and therefore one can take the natural inclusion maps i=1 P ֒→ BT k Q as MU∗-generators of MU∗(BT ). k i k C j k By Quillen’s construction, the Kronecker pairing of Φ([M]T ) and [Qi=1 P → BT ] defines the bordism class of the manifold: k k ji k 2ji+1 k C k < Φ([M]T ), [Y P → BT ] >= [(Y S ) ×T M] ∈ MU∗. i=1 i=1

k i− k 2j 1 k Therefore, Φ([M]T ) is zero if and only if [(Qi=1 S ) ×T M]=0 ∈ MU∗ for all the k C ji k .(inclusions Qi=1 P ֒→ BT for all J =(j1, j2, ..., jk 9 T k On the other hand, by Theorem 3.5, we see the equivariant Chern numbers cω [M]T k = k i− 2j 1 k 0 if and only if the ordinary Chern numbers cω[(Qi=1 S ) ×T M]=0 for all ω and all J =(j1, j2, ..., jk). T k Thus, we conclude that the equivariant Chern number cω [M]=0 for all ω if and only if Φ([M]) = 0. Hence, Theorem 1.5 has been proved.

Together with the above arguments, we conclude: Theorem 3.6. Let M be a closed unitary T k-manifold. Then the following statements are equivalent. U,T k (1) M is null-bordant in Ω∗ . (2) All integral equivariant cohomology Chern numbers of M vanish. k i 2j +1 k (3) For any partition J =(j1, ..., jk) with each ji ≥ 0, ( i=1 S )×T M is null-bordant U Q in Ω∗ . k k ∗ k (4) The fibration π : ET ×T k M −→ BT is null-cobordant in MU (BT ). 3.3. Equivariant unoriented bordism. Our approach above can also be carried out in k k (Z2) O,(Z2) the case of equivariant unoriented bordism. Let N∗ (or Ω∗ ) be the ring formed k by the equivariant bordism classes of all unoriented closed smooth (Z2) -manifolds, and let MO∗(X) be the unoriented (homotopic) cobordism ring of a topological space X, i.e., ∗ l−∗ MO (X) = lim [S ∧ X+,MO(l)]. l−→∞ Then Quillen’s geometric approach on MU ∗(X) can be carried out in the case of MO∗(X), so that the Kronecker pairing ∗ h , i : MO (X) ⊗ MO∗(X) −→ MO∗ =∼ N∗ can be calculated in a geometric way similar to the unitary case as in Subsection 2.1 (cf. [11, D.2.8, D.3.4, Appendix D]), where N∗ is the nonequivariant Thom unoriented bordism ring. In [39], tom Dieck showed that there is also a monomorphism k (Z2) ∗ k ΦR : N∗ −→ MO (B(Z2) ). MO∗(B(Z )k) = lim MO∗(B(Z )k(n)) B(Z )k(n)=(RP n)k It is well-known that 2 ←− 2 where 2 k k and B(Z2) = ∪nB(Z2) (n). Then, applying the finite approximation method yields k (Z2) k N k that for a class [M](Z2) ∈ ∗ , the image ΦR([M](Z2 ) ) is geometrically represented Z k k Z k by the fibration π : E( 2) ×(Z2) M −→ B( 2) . k Theorem 3.7. Let M be a closed smooth (Z2) -manifold. Then the following statements are equivalent. k (Z2) (1) M is null-bordant in N∗ . (2) All equivariant Stiefel–Whitney numbers of M vanish. k i j +1 k (3) For any partition I = (j1, ..., jk) with each ji ≥ 0, (Qi=1 S ) ×(Z2) M is null- bordant in N∗. ∗ Z k k Z k Z k (4) The fibration π : E( 2) ×(Z2) M −→ B( 2) is null-cobordant in MO (B( 2) ). Proof. The proof follows closely that of Theorem 3.6 in a very similar way. We would like to leave it as an exercise to the reader.  10 k Remark 1. Theorem 3.7 tells us that (Z2) -equivariant unoriented bordism is determined by k (Z2) -equivariant Stiefel–Whitney numbers. This result is essentially due to tom Dieck with an argument of involving the Boardman map ∗ k ∗ k B : MO (B(Z2) ) −→ H (B(Z2) ; Z2)⊗H∗(BO; Z2) b (see [39]). Here we exactly replace the use of the Boardman map by the Kronecker pairing. ∗ k It is well-known that as an N∗-algebra, MO (B(Z2) ) is isomorphic to N∗[[v1, ..., vk]], where vi denotes the first cobordism Stiefel–Whitney class of the canonical line bundle k over the i-th factor of B(Z2) for i = 1, ..., k. Thus, in the sense of Buchstaber–Panov– k Ray [10], we can also define (Z2) -equivariant universal genus, which is an N∗-algebra homomorphism, as follows: k (Z2) ΦR : N∗ −→ N∗[[v1, ..., vk]], R ω ω j1 jk by ΦR([M] Z k ) = g (M)v , where v = v ··· v for ω = (j , ..., j ) with ( 2) Pω=(j1,...,jk) ω 1 k 1 k R N R ji ≥ 0, and gω (M) ∈ ∗ with dim gω (M)= |ω| + dim M. R The determination of coefficients gω (M) depends upon the choices of a dual bases k ω ∗ k in MO∗(B(Z2) ) to the basis {v } in MO (B(Z2) ). Buchstaber–Panov–Ray’s approach in the unitary case provides us much more insight on the determination of coefficients R gω (M). In our case, since it was shown in [29, Corollary 6.4] that any real Bott manifold is null-bordant in N∗, we may employ the real Bott manifolds to realize a dual basis Z k ω ∗ Z k R in MO∗(B( 2) ) of the basis {v } in MO (B( 2) ), so that the coefficients gω (M) can R 1 ω ω Z ω be represented by manifolds Gω(M)=(S ) ×(Z2) M, where the action of ( 2) = j1 jk 1 ω 1 j1 1 jk (Z2) ×···× (Z2) on (S ) =(S ) ×···× (S ) is coordinatewise, and on M via the representation (g1,1, ..., g1,j1 ; ...; gk,1, ..., gk,jk ) 7−→ (g1,j1 , ..., gk,jk ). The argument is close to that of the unitary case in [11, §9.2].

REFERENCES [1] J.F. Adams, Stable homotopy and generalised homology, Reprint of the 1974 original, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995. [2] C. Allday and V. Puppe, Cohomological Methods in Transformation Groups, Cambridge Studies in Ad- vanced Mathematics, 32, Cambridge University Press, 1993. [3] M.F. Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200–208. [4] M.F. Atiyah and R. Bott, The Lefschetz fixed-point theorem for elliptic complexes–II, Ann. of Math. 87 (1968), 451–491. [5] M.F. Atiyah and G.B. Segal, The index of elliptic operators–II, Ann. of Math. 87 (1968), 531–545. [6] M.F. Atiyah and I.M. Singer, The index of elliptic operators–I, Ann. of Math. 87 (1968), 484–530. [7] M. Bix and T. tom Dieck, Characteristic numbers of G-manifolds and multiplicative induction. Trans. Amer. Math. Soc. 235 (1978), 331–343. [8] J. M. Boardman, Stable Homotopy Theory: Chapter VI. Mimeographed Notes, University of Warwick, Coventry, UK, 1966. [9] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces, I. Am. J. Math.80 (1959), 458–538. [10] V. Buchstaber, T. Panov and N. Ray, Toric genera. Int. Math. Res. Not. IMRN 2010, no. 16, 3207–3262. [11] V. Buchstaber and T. Panov, Toric Topology, Mathematical Surveys and Monographs, 204. American Mathematical Society, Providence, RI, 2015. [12] G. Comezana, Calculations in complex equivariant bordism. In: J.P. May (ed.), Equivariant Homotopy and Cohomologye theory, CBMS Regional Conference Series in Mathematics 90 (1996), 333–352. [13] P.E. Conner, Seminar on Periodic Maps. Lecture Notes in Mathematics 46. New York: Springer, 1967. [14] P.E. Conner and E.E. Floyd, Differiable periodic maps, Bull. Amer. Math. Soc. 68 (1962), 76–86. 11 [15] P.E. Conner and E.E. Floyd, Differentiable periodic maps, Ergebnisse Math. Grenzgebiete, N. F. Bd. 33, Springer-Verlag, Berlin, 1964. [16] P.E. Conner and E.E. Floyd, Periodic maps which preserve a complex structure, Bull. Amer. Math. Soc. 70 (1964), 574–579. [17] Johannes Ebert, Algebraic independence of generalized MMM-classes. Algebr. Geom. Topol. 11 (2011), no. 1, 69–105. [18] Soren Galatius and Oscar Randal-Williams, Detecting and realising characteristic classes of manifold bundles. arXiv:1304.5571v1 [19] Giansiracusa, Jeffrey and Tillmann, Ulrike, Vanishing of universal characteristic classes for handlebody groups and boundary bundles. J. Homotopy Relat. Struct. 6 (2011), no. 1, 103-112 [20] V. Guillemin, V. Ginzburg and Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions. Appendix J by Maxim Braverman. Mathematical Surveys and Monographs, 98. American Mathe- matical Society, Providence, RI, 2002. [21] B. Hanke, Geometric versus homotopy theoretic equivariant bordism. Math. Ann. 332 (2005), no. 3, 677– 696. [22] A. Hattori, Equivariant characteristic numbers and integrality theorem for unitary T n-manifolds, Tohoku Math. J. (2) 26 (1974), 461–482. [23] S.S.Khare, Characteristic numbers for oriented singular G-bordism, Indian J. Pure Appl. Math. 13 (1982), no. 6, 637–642. [24] S.O.Kochman, Bordism, stable homotopy and Adams spectral sequences. Fields Institute Monographs, 7. American Mathematical Society, Providence, RI, 1996. [25] I.M. Krichever, Obstructions to the existence of S1-actions. Bordisms of ramified coverings. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 40 (1976), 828–844. [26] P.S. Landweber, A survey of bordism and cobordism, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 2, 207–223. [27] C.N. Lee and A.G. Wasserman, Equivariant characteristic numbers. Proceedings of the Second Con- ference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I, pp. 191–216. Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972. [28] P. Löffler, Bordismengruppen unitärer Torusmannigfaltigkeiten. (German) Manuscripta Math. 12 (1974), 307–327. [29] Z. Lü and Q. B. Tan, Small covers and the equivariant bordism classification of 2-torus manifolds, Int. Math. Res. Not. IMRN 2014, no. 24, 6756–6797. [30] Z.LüandQ.B.Tan, Equivariant Chern numbers and the number of fixed points for unitary torus manifolds. Math. Res. Lett. 18 (2011), no. 6, 1319–1325. [31] J. Milnor, On the cobordism ring Ω∗ and a complex analogue. I. Amer. J. Math. 82 (1960), 505–521. [32] S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Soviet Math. Dokl. 1 (1960), 717–720. [33] D. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations. Advances in Mathematics. 7 (1971), 29–56. [34] Dev. P. Sinha, Computations of complex equivariant bordism rings. Amer. J. Math. 123 (2001), no. 4, 577-605. [35] R. E. Stong, Notes on cobordism theory. Mathematical notes 7, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1968. [36] R.E. Stong, Equivariant bordism and Smith theory. IV. Trans. Amer. Math. Soc. 215 (1976), 313–321. [37] R. Thom, Quelques propriét´sglobales des variét´sdifférentiables, Gomm. Math. Helv. 28 (1954), 17–86. [38] T. tom Dieck, Bordism of G-manifolds and integrality theorems. Topology 9 (1970), 345–358. [39] T. tom Dieck, Characteristic numbers of G-manifolds. I. Invent. Math. 13 (1971), 213–224. [40] T. tom Dieck, Characteristic numbers of G-manifolds. II. J. Pure Appl. Algebra 4 (1974), 31–39. [41] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. 72 (1960), 292–311.

SCHOOL OF MATHEMATICAL SCIENCES, FUDAN UNIVERSITY,SHANGHAI, 200433, P.R. CHINA. E-mail address: [email protected] 12 COLLEGE OF INFORMATION TECHNOLOGY,SHANGHAI OCEAN UNIVERSITY, 999HUCHENG HUAN ROAD, 201306, SHANGHAI, P.R. CHINA. E-mail address: [email protected]

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