New York Journal of Mathematics Cobordism Bicycles of Vector Bundles

New York Journal of Mathematics New York J. Math. 26 (2020) 950–1001.

Cobordism bicycles of vector bundles

Shoji Yokura

ABSTRACT. The main ingredient of the algebraic cobordism of M. Levine and h F. Morel is a cobordism cycle of the form (M −→ X; L1, ··· ,Lr) with a proper map h from a smooth variety M and line bundles Li’s over M. In this paper, p we consider a cobordism bicycle of a finite set of line bundles (X ←− V −→s Y ; L1, ··· ,Lr) with a proper map p and a smooth map s and line bundles Li’s over V . We will show that the Grothendieck group Z ∗(X,Y ) of the abelian monoid of the isomorphism classes of cobordism bicycles of finite sets of line bundles satisfies properties similar to those of Fulton-MacPherson’s bivariant theory and also that Z ∗(X,Y ) is a universal one among such abelian groups, i.e., for any abelian group B∗(X,Y ) satisfying the same properties there exists a unique Grothendieck transformation γ : Z ∗(X,Y ) → B∗(X,Y ) preserving the unit.

CONTENTS 1. Introduction 950 2. Fulton–MacPherson’s bivariant theory 955 3. Oriented bivariant theory and a universal oriented bivariant theory 959 4. Cobordism bicycles of vector bundles 963 5. Cobordism bicycles of ﬁnite tuples of line bundles 976 References 999

1. Introduction V. Voevodsky first introduced algebraic cobordism or higher algebraic cobordism MGL∗,∗(X) in the context of motivic homotopy theory and used it in his proof of the Milnor conjecture [26, 28, 27]. Later, in an attempt to understand MGL∗,∗(X) better, M. Levine and F. Morel [20] constructed another algebraic cobordism Ω∗(X) in terms of what they call a cobordism cycle (which is of the h form [V −→ X; L1,L2, ··· ,Lr] with line bundles Li’s over V which is smooth) and some relations on these cobordism cycles, as the universal oriented cohomology theory. To be a bit more precise, using cobordism cycles they first defined an oriented Borel–Moore functor Z∗ with products satisfying twelve conditions

Received June 19, 2019. 2010 Mathematics Subject Classiﬁcation. 55N35, 55N22, 14C17, 14C40, 14F99, 19E99. Key words and phrases. (co)bordism, algebraic cobordism, algebraic cobordism of bundles, correspondence.

ISSN 1076-9803/2020 950 COBORDISMBICYCLESOFVECTORBUNDLES 951

(D1) - (D4) and (A1) - (A8), and then deﬁned an oriented Borel–Moore functor with products of geometric type by further imposing on the functor Z∗ the relations R corresponding to three axioms (Dim) (dimension axiom), (Sect) (section axiom) and (FGL) (formal group law axiom), which correspond to “of geometric type”. The functor Z∗/R is nothing but Levine–Morel’s algebraic cobordism Ω∗. ∼ 2∗,∗ In [19], M. Levine showed that there is an isomorphism Ω∗(X) = MGL (X) for smooth X. f In [12], W. Fulton and R. MacPherson have introduced bivariant theory B(X −→ Y ) with an aim to deal with Riemann–Roch type theorems for singular spaces −∗ πX and to unify them. The extreme cases B∗(X) := B (X −−→ pt) becomes a ∗ ∗ idX covariant functor and B (X) := B (X −−→ X) becomes a contravariant functor. f In this sense, B(X −→ Y ) is called a bivariant theory. In [29] (cf. [30]), the author introduced an oriented bivariant theory and a universal oriented bivariant f theory in order to construct a bivariant-theoretic version Ω∗(X −→ Y ) of Levine– π Morel’s algebraic cobordism so that the covariant part Ω−∗(X −−→X pt) becomes isomorphic to Levine–Morel’s algebraic cobordism Ω∗(X). prop f Our universal oriented bivariant theory OMsm (X −→ Y ) is deﬁned to be p the Grothendieck group of the abelian monoid of the isomorphism classes [V −→ X; L1,L2, ··· ,Lr] such that: (1) h : V → X is a proper map, (2) the composite f ◦ h : V → Y is a smooth map. (Note that this requirement implies that if the target Y is the point pt, then the source V has to be h smooth, thus [V −→ X; L1,L2, ··· ,Lr] becomes a cobordism cycle in the sense of Levine–Morel.)

Here for the monoid we consider the following addition

p1 p2 0 0 [V1 −→ X; L1, ··· ,Lr] + [V2 −→ X; L1, ··· ,Lr] (1.1) p1tp2 0 0 := [V1 t V2 −−−→ X; L1 t L1, ··· ,Lr t Lr]

0 where t is the disjoint union and Li t Li is a line bundle over V1 t V2 such that 0 0 0 prop f (Li t Li)|V1 = Li and (Li t Li)|V2 = Li. In other words, OMsm (X −→ Y ) is the p free abelian group generated by the isomorphism classes [V −→ X; L1,L2, ··· ,Lr] modulo the additivity relation (1.1).1

1We remark that in [29] we consider the free abelian group generated by the isomorphism p classes [V −→ X; L1,L2, ··· ,Lr], however the results in [29] still hold even if we consider the Grothendieck group by modding it out by the additivity relation (1.1). 952 SHOJI YOKURA

forget the map f {L1, ··· Lr} ======⇒ {L1, ··· Lr}

V V h f◦h h f◦h

X y %/ Y XYy % f (1.2) If we forget or ignore the given map f in the left-hand-side diagram above, then we get the right-hand-side diagram above, which is

h f◦h a correspondence X ←− V −−→ Y with a ﬁnite set of line bundles {L1, ··· ,Lr}, where h : V → X is a proper map and f ◦ h : V → Y is a smooth map. Such a correspondence (or sometimes called a span or a roof) can be considered for any pair (X,Y ) of varieties X and Y :

{L1, ··· Lr} E

V V p s p s x & XY,y % XY with a proper map p and a smooth map s. In the right-hand-side diagram E is one vector bundle over V , not necessarily a line bundle. These two correspondences p s p s are denoted by (X ←− V −→ Y ; L1, ··· Lr) and (X ←− V −→ Y ; E) respectively. It turns out that such a correspondence has been already studied in C∗-algebra, in particular for Kasparov’s KK-theory KK(X,Y ) ([17]), another kind of bivariant theory which has been studied by many people in operator theory. For example, in b f [9], (cf. [5] and [10]) A. Connes and G. Skandalis consider (X ←− M −→ Y ; ξ) with b a proper map, f a smooth K-oriented map and ξ a vector bundle over M. In [6], P. Baum and J. Block consider such a correspondence for singular spaces with a group action on it and call such a correspondence an equivariant bicycle. So we shall call the above correspondence a cobordism bicycle of vector bundles. p In our previous paper [31], we consider the above cobordism bicycle (X ←− s V −→ Y ; E) of vector bundles as a morphism from X to Y and furthermore consider the enriched category of such cobordism bicycles of vector bundles. Then we extend Baum–Fulton–MacPherson’s Riemann–Roch (or Todd class) transformation τ : G0(−) → H∗(−) ⊗ Q (see [7]) to this enriched category. If we consider the above (1.2), it is quite natural or reasonable to think that there must be a connection or a relation between the above two kinds of bivariant COBORDISMBICYCLESOFVECTORBUNDLES 953 theories, Fulton–MacPherson’s bivariant theory (in topology) and Kasparov’s bivariant theory (in operator theory). So, as an intermediate theory between these two theories we consider the free abelian group Z ∗(X,Y ) generated by isomorphism classes of cobordism bicycles of vector bundles and ﬁnite sets of line bundles modulo the additive relation like (1.1):

p1 s1 p2 s2 p1tp2 s1ts2 [X ←− V1 −→ Y ; E1]+[X ←− V2 −→ Y ; E2] := [X ←−−− V2 −−−→ Y ; E1tE2],

p1 s1 p2 s2 0 0 [X ←− V1 −→ Y ; L1, ··· ,Lr] + [X ←− V2 −→ Y ; L1, ··· ,Lr] p1tp2 s1ts2 0 0 := [X ←−−− V2 −−−→ Y ; L1 t L1, ··· ,Lr t Lr]. For example, we can show the following:

Theorem 1.3. For a pair (X,Y ) the Grothendieck group Z ∗(X,Y ) of the abelian monoid of the isomorphism classes of cobordism bicycles of finite sets of line bundles satisfies the following (similar to those of Fulton–MacPherson’s bivariant theory): (1) it is equipped with the following three operations (1) (product) • : Z i(X,Y ) ⊗ Z j(Y,Z) → Z i+j(X,Z) (2) (Pushforward) 0 i i 0 (a) For a proper map f : X → X , f∗ : Z (X,Y ) → Z (X ,Y ). 0 2 i i+dim g 0 (b) For a smooth map g : Y → Y , ∗g : Z (X,Y ) → Z (X,Y ). (3) (Pullback) (a) For a smooth map f : X0 → X, f ∗ : Z i(X,Y ) → Z i+dim f (X0,Y ). (b) For a proper map g : Y 0 → Y , ∗g3 : Z i(X,Y ) → Z i(X,Y 0). (2) the three operations satisfies the following nine properties:

(A1) Product is associative. (A2) Pushforward is functorial. (A2)’ Proper pushforward and smooth pushforward commute. (A3) Pullback is functorial. (A3)’ Proper pullback and smooth pullback commute. (A12) Product and pushforward commute. (A13) Product and pullback commute. (A23) Pushforward and pullback commute. (A123) Projection formula. ∗ 0 (3) Z has units, i.e., there is an element 1X ∈ Z (X,X) such that 1X • α = α ∗ ∗ for any element α ∈ Z (X,Y ) and β • 1X = β for any element β ∈ Z (Y,X). (4) Z ∗ satisﬁes PPPU (Pushforward-Product Property for units) and PPU (Pull- back Property for units). (For the details of these properties, see Lemma 5.9 and Lemma 5.14.)

2 For this unusual notation ∗g instead of g∗ see §4 3For this unusual notation ∗g instead of g∗ see §4 954 SHOJI YOKURA

(5) Z ∗ is equipped with the Chern class operators: for a line bundle L over X and a line bundle M over Y i i+1 i i+1 c1(L)• : Z (X,Y ) → Z (X,Y ), •c1(M): Z (X,Y ) → Z (X,Y ) which satisfy the properties listed in Lemma 5.11. We do not have a reasonable name for this naive theory Z ∗(X,Y ) satisfying those properties above, so in this paper we call it a “bi-variant” theory. Deﬁnition 1.4. Let B, B0 be two bi-variant theories on a category V .A Grothen- dieck transformation γ : B → B0 is a collection of homomorphisms B(X,Y ) → B0(X,Y ) for a pair (X,Y ) in the category V , which preserves the above three basic operations and the Chern class operator:

(1) γ(α •B β) = γ(α) •B0 γ(β), (2) γ(f∗α) = f∗γ(α) and γ(α ∗g) = γ(α) ∗g , (3) γ(g∗α) = g∗γ(α) and γ(α ∗f) = γ(α) ∗f (4) γ(c1(L) • α) = c1(L) • γ(α) and γ(α • c1(M)) = γ(α) • c1(M). We show the following theorem. Theorem 1.5. The above Z ∗(−, −) is the universal one among bi-variant theories in the sense that given any bi-variant theory B∗(−, −), there exists a unique Grothendieck transformation ∗ ∗ γB : Z (−, −) → B (−, −) such that γB(11V ) = 1V ∈ B(V,V ) for any variety V . Remark 1.6. In [1], T. Annala has succeeded in constructing what he calls the bi- f variant derived algebraic cobrodism Ω∗(X −→ Y ), a bivariant theoretic analogue of Levine–Morel’s algebraic cobordism Ω∗(X) (which the author has been trying to aim at), using the construction of Lowrey-Schurg’s¨ derived algebraic cobordism dΩ∗(X) [23] in derived algebraic geometry and the author’s construction of the universal bivariant theory. Roughly speaking, in [1], Annala considers the bivari- prop f ant theory OMqusm(X −→ Y ) for the category of derived algebraic schemes in derived algebraic geometry, where qusm refers to quai-smooth morphisms, and f prop f furthermore imposes some relations R(X −→ Y ) on OMqusm(X −→ Y ) to ob- pro f f tain its quotient OMqusm(X −→ Y )/R(X −→ Y ) , which is the bivariant derived f algebraic cobordism Ω∗(X −→ Y ). The “forget” map deﬁned in (1.2) gives rise to the following canonical homomorphism prop f ∗ f : OMsm (X −→ Y ) → Z (X,Y ) p p f◦p deﬁned by f([V −→ Y ; L1, ··· ,Lr]) := [X ←− V −−→ Y ; L1, ··· ,Lr]. This “forget” map is compatible with the bivariant product •, the bivariant pushforward and the Chern class operator, but not necessarily with the pullback. As to correspondences, D. Gaitsgory and N. Rozenblyum study correspondences in derived COBORDISMBICYCLESOFVECTORBUNDLES 955 algebraic geometry intensively in their recent book [13] (cf. [14]) . For example they consider the category Corr(C)vert,horiz for a category C equipped with two classes of morphisms vert and horiz (both closed under composition) such that

(1) the objets of Corr(C)vert,horiz are the same as those of C and (2) the morphisms are correspondences, i.e., a morphism from c0 to c1 is a correspondence (drawn as follows in [13]):

g c0,1 / c0

f c1 where f is vert and g is horiz. If we use our notation, Corr(C)vert,horiz can horiz g f be denoted by Corr(C)vert and the above diagram is c0 ←− c0,1 −→ c1. So, in this way of thinking of two kinds of bivariant theories, it remains to see whether one could get a “correspondence” version of Annala’s bivariant derived algebraic cobordism, i.e., whether one could consider some reasonable relations R(X,Y ) prop ∗ on Zqusm(X,Y ) which is a derived algebraic geometric version of Z (X,Y ) with smooth morphism being replaced by quasi-smooth morphism, such that the following diagram commutes:

prop f f prop OMqusm(X −→ Y ) / Zqusm(X,Y )

π π prop f prop (X−→Y ) f Z (X,Y ) OMqusm =: Ω∗(X −→ Y ) / Z ?(X,Y ) := qusm . f f R(X,Y ) R(X−→Y )

We hope to be able to treat this problem in a different paper. It would be nice that there were some relation between Annala’s bivariant derived algebraic cobordism f Ω∗(X −→ Y ) and Kasparov’s bivariant KK-theory KK(X,Y ) via the “forget” f map f :Ω∗(X −→ Y ) → KK(X,Y ) for certain reasonable maps f : X → Y .

2. Fulton–MacPherson’s bivariant theory We make a quick review of Fulton–MacPherson’s bivariant theory [12] (also see [11]) (cf. a universal bivariant theory [29, 30]). Let V be a category which has a final object pt and on which the fiber product or fiber square is well-defined. Also we consider a class of maps, called “confined maps” (e.g., proper maps, projective maps, in algebraic geometry), which are closed under composition and base change and contain all the identity maps, and a class of fiber squares, called “independent squares” (or “confined squares”, e.g., “Tor-independent” in algebraic geometry, a fiber square with some extra conditions required on morphisms of the square), which satisfy the following: 956 SHOJI YOKURA

(i) if the two inside squares in X0 −−−−→ X h00 0 0   00 h 0 g 0  X −−−−→ X −−−−→ X f y yf    0 f 00 f 0 f or Y −−−−→ Y y y y h0 00 0   Y −−−−→ Y −−−−→ Y g0 g h g y y Z0 −−−−→ Z h are independent, then the outside square is also independent, (ii) any square of the following forms are independent:

idX f X / X X / Y

f f idX idY Y / YX / Y idX f where f : X → Y is any morphism. A bivariant theory B on a category V with values in the category of graded f abelian groups is an assignment to each morphism X −→ Y in the category V a f graded abelian group (in most cases we ignore the grading ) B(X −→ Y ) which is f equipped with the following three basic operations. The i-th component of B(X −→ i f Y ), i ∈ Z, is denoted by B (X −→ Y ). (1) Product: For morphisms f : X → Y and g : Y → Z, the product operation

i f j g i+j gf • : B (X −→ Y ) ⊗ B (Y −→ Z) → B (X −→ Z) is deﬁned. (2) Pushforward: For morphisms f : X → Y and g : Y → Z with f conﬁned, the pushforward operation

i gf i g f∗ : B (X −→ Z) → B (Y −→ Z) is deﬁned. g0 X0 −−−−→ X   0  (3) Pullback : For an independent square f y yf Y 0 −−−−→ Y, g the pullback operation

0 ∗ i f i 0 f 0 g : B (X −→ Y ) → B (X −→ Y ) is deﬁned. COBORDISMBICYCLESOFVECTORBUNDLES 957

These three operations are required to satisfy the following seven compatibility axioms ([12, Part I, §2.2]): f g h f (A1) Product is associative: for X −→ Y −→ Z −→ W with α ∈ B(X −→ g h Y ), β ∈ B(Y −→ Z), γ ∈ B(Z −→ W ),

(α • β) • γ = α • (β • γ).

f g h (A2) Pushforward is functorial : for X −→ Y −→ Z −→ W with f and g h◦g◦f conﬁned and α ∈ B(X −−−−→ W )

(g ◦ f)∗(α) = g∗(f∗(α)).

(A3) Pullback is functorial: given independent squares

0 g0 X00 −−−−→h X0 −−−−→ X     00  0  yf yf yf Y 00 −−−−→ Y 0 −−−−→ Y h g

(g ◦ h)∗ = h∗ ◦ g∗.

f g h (A12) Product and pushforward commute: for X −→ Y −→ Z −→ W with f g◦f h conﬁned and α ∈ B(X −−→ Z), β ∈ B(Z −→ W ),

f∗(α • β) = f∗(α) • β.

(A13) Product and pullback commute: given independent squares

00 X0 −−−−→h X   0  f y yf 0 Y 0 −−−−→h Y   0 g g y y Z0 −−−−→ Z h

f g with α ∈ B(X −→ Y ), β ∈ B(Y −→ Z),

h∗(α • β) = h0∗(α) • h∗(β). 958 SHOJI YOKURA

(A23) Pushforward and pullback commute: given independent squares 00 X0 −−−−→h X   0  f y yf 0 Y 0 −−−−→h Y   0 g g y y Z0 −−−−→ Z h g◦f with f conﬁned and α ∈ B(X −−→ Z), 0 ∗ ∗ f∗(h (α)) = h (f∗(α)).

(A123) Projection formula: given an independent square with g conﬁned and f 0 h◦g α ∈ B(X −→ Y ), β ∈ B(Y −−→ Z) g0 X0 −−−−→ X   0  f y yf Y 0 −−−−→ Y −−−−→ Z g h

f 0 h◦g and α ∈ B(X −→ Y ), β ∈ B(Y −−→ Z), 0 ∗ g∗(g (α) • β) = α • g∗(β). We also assume that B has units: 0 idX Units: B has units, i.e., there is an element 1X ∈ B (X −−→ X) such that α • 1X = α for all morphisms W → X and all α ∈ B(W → X), such that 1X • β = β for all morphisms X → Y and all β ∈ B(X → Y ), and such that ∗ 0 g 1X = 1X0 for all g : X → X. Commutativity: B is called commutative if whenever both g0 f 0 W −−−−→ X W −−−−→ Y     0  0 g f y yf and g y y Y −−−−→ Z X −−−−→ Z g g f g are independent squares with α ∈ B(X −→ Z) and β ∈ B(Y −→ Z), g∗(α) • β = f ∗(β) • α. 0 Let B, B be two bivariant theories on a category V .A Grothendieck transforma- 0 0 tion from B to B , γ : B → B is a collection of homomorphisms B(X → Y ) → 0 B (X → Y ) for a morphism X → Y in the category V , which preserves the above three basic operations:

(1) γ(α •B β) = γ(α) •B0 γ(β), COBORDISMBICYCLESOFVECTORBUNDLES 959

(2) γ(f∗α) = f∗γ(α), and (3) γ(g∗α) = g∗γ(α). A bivariant theory unifies both a covariant theory and a contravariant theory in the following sense: B∗(X) := B(X → pt) becomes a covariant functor for confined morphisms and ∗ id B (X) := B(X −→ X) becomes a contravariant functor for any morphisms. 0 A Grothendieck transformation γ : B → B induces natural transformations γ∗ : 0 ∗ ∗ 0∗ B∗ → B∗ and γ : B → B . −i j Definition 2.1. As to the grading, Bi(X) := B (X → pt) and B (X) := j id B (X −→ X). Definition 2.2. ([12, Part I, §2.6.2 Definition]) Let S be a class of maps in V , which is closed under compositions and containing all identity maps. Suppose that f to each f : X → Y in S there is assigned an element θ(f) ∈ B(X −→ Y ) satisfying that (i) θ(g ◦ f) = θ(f) • θ(g) for all f : X → Y , g : Y → Z ∈ S and ∗ ∗ idX (ii) θ(idX ) = 1X for all X with 1X ∈ B (X) := B (X −−→ X) the unit. Then θ(f) is called an orientation of f. (In [12, Part I, §2.6.2 Definition], it is called a canonical orientation of f, but in this paper it shall be simply called an orientation.) Definition 2.3. Let S be another class of maps called “specialized maps” (e.g., smooth maps in algebraic geometry) in V , which is closed under composition, closed under base change and containing all identity maps. Let B be a bivariant theory. If S has an orientation θ for B and it satisfies that for an independent square with f ∈ S g0 X0 −−−−→ X   0  f y yf Y 0 −−−−→ Y g the following condition holds: θ(f 0) = g∗θ(f), (which means that the orientation θ is preserved by the pullback operation), then we call θ a stable orientation and say that S is stably B-oriented. 3. Oriented bivariant theory and a universal oriented bivariant theory Levine–Morel’s algebraic cobordism is the universal one among the so-called oriented Borel–Moore functors with products for algebraic schemes. Here “oriented” means that the given Borel–Moore functor H∗ is equipped with the endomorphism c˜1(L): H∗(X) → H∗(X) for a line bundle L over the scheme X. Motivated by this “orientation” (which is different from the one given in Definition 2.2, but we still call this “orientation” using a different symbol so that the reader 960 SHOJI YOKURA will not be confused with terminologies), in [29, §4] we introduce an orientation to bivariant theories for any category, using the notion of fibered categories in abstract category theory (e.g, see [25]) and such a bivariant theory equipped with such an orientation (Chern class operator) is called an oriented bivariant theory.

Definition 3.1. ([29, Definition 4.2]) (an oriented bivariant theory) Let B be a bivariant theory on a category V . (1) For a fiber-object L over X, the “operator” on B associated to L, denoted by φ(L), is defined to be an endomorphism

f f φ(L): B(X −→ Y ) → B(X −→ Y ) which satisﬁes the following properties: (O-1) identity: If L and L0 are two ﬁber-objects over X and isomorphic (i.e., if f : L → X and f 0 : L0 → X, then there exists an isomorphism i : L → L0 such that f = f 0 ◦ i) , then we have

0 f f φ(L) = φ(L ): B(X −→ Y ) → B(X −→ Y ). (O-2) commutativity: Let L and L0 be two ﬁber-objects over X, then we have

0 0 f f φ(L) ◦ φ(L ) = φ(L ) ◦ φ(L): B(X −→ Y ) → B(X −→ Y ). (O-3) compatibility with product: For morphisms f : X → Y and f g g : Y → Z, α ∈ B(X −→ Y ) and β ∈ B(Y −→ Z), a fiber-object L over X and a fiber-object M over Y , we have φ(L)(α • β) = φ(L)(α) • β, φ(f ∗M)(α • β) = α • φ(M)(β). (O-4) compatibility with pushforward: For a confined morphism f : X → Y and a fiber-object M over Y we have ∗ f∗ (φ(f M)(α)) = φ(M)(f∗α). (O-5) compatibility with pullback: For an independent square and a fiber-object L over X

g0 X0 −−−−→ X   0  f y yf Y 0 −−−−→ Y g we have g∗ (φ(L)(α)) = φ(g0∗L)(g∗α). The above operator is called an “orientation” and a bivariant theory equipped with such an orientation is called an oriented bivariant theory, denoted by OB. COBORDISMBICYCLESOFVECTORBUNDLES 961

(2) An oriented Grothendieck transformation between two oriented bivariant theories is a Grothendieck transformation which preserves or is compatible with the operator, i.e., for two oriented bivariant theories OB with an 0 0 orientation φ and OB with an orientation φ the following diagram commutes. f φ(L) f OB(X −→ Y ) −−−−→ OB(X −→ Y )   γ γ y y 0 f 0 f OB (X −→ Y ) −−−−→ OB (X −→ Y ). φ0(L)

Theorem 3.2. ([29, Theorem 4.6]) (A universal oriented bivariant theory) Let V be a category with a class C of confined morphisms, a class of independent squares, a class S of specialized morphisms and L a fibered category over V . We define

C f OMS (X −→ Y ) to be the free abelian group generated by the set of isomorphism classes of cobordism cycles over X h [V −→ X; L1,L2, ··· ,Lr] such that h ∈ C , f ◦ h : W → Y ∈ S and Li a fiber-object over V . C (1) The association OMS becomes an oriented bivariant theory if the four operations are defined as follows: (a) Orientation Φ: For a morphism f : X → Y and a fiber-object L over X, the operator

C f C f Φ(L): OMS (X −→ Y ) → OMS (X −→ Y ) is deﬁned by

h h ∗ Φ(L)([V −→ X; L1,L2, ··· ,Lr]) := [V −→ X; L1,L2, ··· ,Lr, h L]. and extended linearly. (b) Product: For morphisms f : X → Y and g : Y → Z, the product operation

C f C g C gf • : OMS (X −→ Y ) ⊗ OMS (Y −→ Z) → OMS (X −→ Z) is deﬁned as follows: The product is deﬁned by

h k [V −→ X; L1, ··· ,Lr] • [W −→ Y ; M1, ··· ,Ms] 0 h◦k00 00∗ 00∗ 0 0 ∗ 0 0 ∗ := [V −−−→ X; k L1, ··· , k Lr, (f ◦ h ) M1, ··· , (f ◦ h ) Ms] 962 SHOJI YOKURA

and extended bilinearly. Here we consider the following ﬁber squares

0 f 0 V 0 −−−−→h X0 −−−−→ W    00 0  k y k y ky V −−−−→ X −−−−→ Y −−−−→ Z. h f g (c) Pushforward: For morphisms f : X → Y and g : Y → Z with f conﬁned, the pushforward operation

C gf C g f∗ : OMS (X −→ Z) → OMS (Y −→ Z) is deﬁned by

h f◦h f∗ [V −→ X; L1, ··· ,Lr] := [V −−→ Y ; L1, ··· ,Lr] and extended linearly. (d) Pullback: For an independent square g0 X0 −−−−→ X   0  f y yf Y 0 −−−−→ Y, g the pullback operation

0 ∗ C f C 0 f 0 g : OMS (X −→ Y ) → OMS (X −→ Y ) is deﬁned by

∗ h 0 h0 0 00∗ 00∗ g [V −→ X; L1, ··· ,Lr] := [V −→ X ; g L1, ··· , g Lr] and extended linearly, where we consider the following ﬁber squares:

g00 V 0 −−−−→ V   0  h y yh g0 X0 −−−−→ X   0  f y yf Y 0 −−−−→ Y. g (2) Let OBT be a class of oriented bivariant theories OB on the same category V with a class C of conﬁned morphisms, a class of independent squares, a class S of specialized morphisms and a ﬁbered category L over V . Let S be stably OB-oriented for any oriented bivariant theory OB ∈ OBT . Then, for each oriented bivariant theory OB ∈ OBT with COBORDISMBICYCLESOFVECTORBUNDLES 963

an orientation φ, there exists a unique oriented Grothendieck transformation C γOB : OMS → OB

C f such that for any f : X → Y ∈ S the homomorphism γOB : OMS (X −→ f Y ) → OB(X −→ Y ) satisﬁes the normalization condition that

idX γOB([X −−→ X; L1, ··· ,Lr]) = φ(L1) ◦ · · · ◦ φ(Lr)(θOB(f)). In this paper, we consider the category V of complex algebraic varieties or schemes and we consider proper morphisms for the class C of confined maps, smooth morphisms for the class S of specialized morphisms, fiber squares for independent squares, and line bundles for a fibered category L over V . So, C f prop f OMS (X −→ Y ) shall be denote by OMsm (X −→ Y ). For a smooth morphism f : X → Y ,

idX prop f θ(f) := [X −−→ X] ∈ OMsm (X −→ Y ) is clearly a stable orientation. As mentioned in the introduction, we can consider prop f the above free abelian group OMsm (X −→ Y ) modulo the additive relation (1.1), i.e., the Grothendieck group of the monoid of the isomorphism classes of cobordism bicycles of ﬁnite sets of line bundles, which is denoted by the same notation prop f OMsm (X −→ Y ).

4. Cobordism bicycles of vector bundles In this section, we consider extending the notion of algebraic cobordism of vector bundles due to Y.-P. Lee and R. Pandharipande [18] (cf. [22]) to correspondences. p s Deﬁnition 4.1. Let X ←− V −→ Y be a correspondence (sometimes called a span or a roof ) such that p : V → X is a proper map and s : V → Y be a smooth map, and let E be a complex vector bundle. Then p (X ←− V −→s Y ; E) is called a cobordism bicycle of a vector bundle.

p s Remark 4.2. The above correspondence X ←− V −→ Y shall be called a proper- smooth correspondence, abusing words. Mimicking naming used in [6], [20] and [18], the above proper-smooth correspondence equipped with a vector bundle is simply also named “cobordism bicycle of a vector bundle”.

p s p0 s0 Deﬁnition 4.3. Let (X ←− V −→ Y ; E) and (X ←− V 0 −→ Y ; E0) be two cobordism bicycles of vector bundles of the same rank. If there exists an isomorphism h : V =∼ V 0 such that 964 SHOJI YOKURA

p s p0 s0 (1) (X ←− V −→ Y ) =∼ (X ←− V 0 −→ Y ) as correspondences, i.e., the following diagrams commute: V p s ~ XYh ` >

0 0 p s V 0 (2) E =∼ h∗E0, then they are called isomorphic and denoted by p s p0 s0 (X ←− V −→ Y ; E) =∼ (X ←− V 0 −→ Y ; E0). p s The isomorphism class of a cobordism bicycle of a vector bundle (X ←− V −→ p s Y ; E) is denoted by [X ←− V −→ Y ; E], which is still called a cobordism bicycle of a vector bundle. For a ﬁxed rank r for vector bundles, the set of isomorphism classes of cobordism bicycles of vector bundles for a pair (X,Y ) becomes a commutative monoid by the disjoint sum:

p1 s1 p2 s2 [X ←− V1 −→ Y ; E1] + [X ←−V2 −→ Y ; E2]

p1+p2 s1+s2 := [X ←−−−− V1 t V2 −−−−→ Y ; E1 + E2], where E1 +E2 is a vector bundle such that (E1 +E2)|V1 = E1 and (E1 +E2)|V2 = p s E2. This monoid is denoted by Mr(X,Y ) and another grading of [X ←− V −→ Y ; E] is defined by the relative dimension of the smooth map s, denoted by dim s, p s thus by double grading, [X ←− V −→ Y ; E] ∈ Mn,r(X,Y ) means that n = dim s and r = rank E. The group completion of this monoid, i.e., the Grothendieck + group, is denoted by Mn,r(X,Y ) . We use this notation, mimicking [20]. + L + Remark 4.4. For a fixed rank r, M∗,r(X,Y ) = Mn,r(X,Y ) is a graded abelian group. + + Remark 4.5. When Y = pt a point, Mn,r(X, pt) is nothing but Mn,r(X) considered in Lee–Pandharipande [18]. In this sense, when X = pt a point, + Mn,r(pt, Y ) is a new object to be investigated. Definition 4.6 (product of cobordism bicycles). For three varieties X,Y,Z, we define the following two kinds of product •⊕ and •⊗: (1) (by the Whitney sum ⊕) + + + •⊕ : Mm,r(X,Y ) ⊗ Mn,k(Y,Z) → Mm+n,r+k(X,Z) ;

p s q t [X ←− V −→ Y ; E] •⊕ [Y ←−W −→ Z; F ] p s q s ∗ ∗ := [(X ←− V −→ Y ) ◦ (Y ←− W −→ Z); qe E ⊕ se F ], COBORDISMBICYCLESOFVECTORBUNDLES 965

(2) (by the tensor product ⊗) + + + •⊗ : Mm,r(X,Y ) ⊗ Mn,k(Y,Z) → Mm+n,rk(X,Z) ;

p s q t [X ←− V −→ Y ; E] •⊗ [Y ←−W −→ Z; F ] p s q s ∗ ∗ := [(X ←− V −→ Y ) ◦ (Y ←− W −→ Z); qe E ⊗ se F ]. Here we consider the following commutative diagram

∗ ∗ ∗ ∗ qe E ⊕ se F or qe E ⊗ se F

E V ×Y W F q e se V u ) W p s q t ~ XYZ) u

Lemma 4.7. The products •⊕ and •⊗ are both bilinear. + Remark 4.8. M∗,∗(X,X) is a double graded commutative ring with respect to both products •⊕ and •⊗.

Remark 4.9. We consider the above product •⊕ for Y = Z = pt a point. Since + + + + Mn,r(X, pt) = Mn,r(X) and Mn,r(pt, pt) = Mn,r(pt) , we have + + + •⊕ : Mm,r(X) ⊗ Mn,k(pt) → Mm+n,r+k(X) .

p s q t [X ←− V −→ pt; E] •⊕ [pt ←− W −→ pt; F ] p s q s ∗ ∗ = [(X ←− V −→ pt) ◦ (pt ←− W −→ pt); (pr1) E ⊕ (pr2) F ], which is rewritten as follows, by using the notations used in [18]:

p p◦pr1 ∗ ∗ [V −→ X,E] •⊕ [W ; F ] = [V × W −−−→ X;(pr1) E ⊕ (pr2) F ].

∗ ∗ (pr1) E ⊕ (pr2) F

E V × W F pr1 pr2 V w ' W p s q t X' ptw pt 966 SHOJI YOKURA

In [18, §0.8] they deﬁne the following map (which turns out to be isomorphic [18, Theorem 3])

γX : ω∗(X) ⊗ω∗(pt) ω∗,r(pt) → ω∗,r(X) of ω∗(pt)-modules by

f λ r−`(λ) M γX [Y −→ X] ⊗ [P , O ⊕ Lm] m∈λ

f◦pY M := [Y × λ −−−→ X, r−`(λ) ⊕ p∗ L ]. P O Pλ m m∈λ

In fact, this map is nothing but our product •⊕ at least at the level of M∗,∗:

•⊕ : M∗,0(X, pt) ⊗ M∗,r(pt, pt) → M∗,r(X, pt).

Note that M∗,0(X, pt) = M∗,0(X), M∗,r(pt, pt) = M∗,r(pt) and M∗,r(X, pt) = M∗,r(X). Using our notation, we have

f f s • [Y −→ X] = [X ←− Y −→ pt], λ r−`(λ) L λ r−`(λ) L • [P , O ⊕ m∈λ Lm] = [pt ←− P −→ pt; O ⊕ m∈λ Lm], λ f◦pY r−`(λ) L ∗ f◦pY λ • [Y × −−−→ X, O ⊕ p λ Lm] = [X ←−−− Y × −→ P m∈λ P P r−`(λ) L ∗ pt; O ⊕ p λ Lm]. m∈λ P Our product •⊕ gives us

f s λ r−`(λ) M [X ←− Y −→ pt]•⊕[pt ←− P −→ pt; O ⊕ Lm] m∈λ

f◦pY M = [X ←−−− Y × λ −→ pt; r−`(λ) ⊕ p∗ L ]. P O Pλ m m∈λ

Indeed, for this product, we consider the following diagram:

∗ r−`(λ) L p λ (O ⊕ Lm) P m∈λ

* λ r−`(λ) L Y × P O ⊕ m∈λ Lm p p λ Y P Y v * Pλ f

X( ptt ' pt COBORDISMBICYCLESOFVECTORBUNDLES 967

∗ r−`(λ) L r−`(λ) L ∗ r−`(λ) Note that p λ (O ⊕ Lm) = O ⊕ p λ Lm since O P m∈λ m∈λ P is a trivial bundle. Therefore, we can see that

f λ r−`(λ) M γX [Y −→ X] ⊗ [P ,O ⊕ Lm] m∈λ f s λ r−`(λ) M = [X ←− Y −→ pt] •⊕ [pt ←− P −→ pt; O ⊕ Lm]. m∈λ

p s + Remark 4.10. Let [X ←− V −→ Y ; E] ∈ Mm,r(X,Y )⊕, and let

p p idV + (1) [X ←− V ] := [X ←− V −−→ V ] ∈ M0,0(X,V )⊕, idV idV + (2) [V ; E] := [V ←−− V −−→ V ; E] ∈ M0,r(V,V )⊕, s idV s + (3) [V −→ Y ] := [V ←−− V −→ Y ] ∈ Mm,0(V,Y )⊕. p s p s Then we have [X ←− V −→ Y ; E] = [X ←− V ] •⊕ [V ; E] •⊕ [V −→ Y ]. Now, we deﬁne pushforward and pullback of cobordism bicycles of vector bundles:

Definition 4.11. (1) (Pushforward) (a) For a proper map f : X → X0, the (proper) pushforward acting on the p + first factor X ←− V , in a usual way denoted by f∗ : Mm,r(X,Y ) → 0 + Mm,r(X ,Y ) , is defined by

p s 0 f◦p s f∗([X ←− V −→ Y ; E]) := [X ←−− V −→ Y ; E]. (b) For a smooth map g : Y → Y 0, the (smooth) pushforward acting s on the second factor V −→ Y , in an unusual way denoted by ∗g : + 0 + Mm,r(X,Y ) → Mm+dim g,r(X,Y ) , is deﬁned by

p s p g◦s 0 ([X ←− V −→ Y ; E]) ∗g := [X ←− V −−→ Y ; E]. p s Here we emphasize that ∗g is written on the right side of ([X ←− V −→ Y ; E]) not on the left side.4 (Note that m = dim s and dim(g ◦ s) = dim s + dim g = m + dim g.) (2) (Pullback) (a) For a smooth map f : X0 → X, the (smooth) pullback acting on the p ∗ + ﬁrst factor X ←− V , in a usual way denoted byf : Mm,r(X,Y ) → 0 + Mm+dim f,r(X ,Y ) is deﬁned by

0 0 ∗ p s 0 p 0 s◦f 0 ∗ f ([X ←− V −→ Y ; E]) := [X ←− X ×X V −−→ Y ;(f ) E].

4 ∗ For pushforward the notation ∗g and writing it on the right side and for pullback the notation g and writing it on the right side were suggested by the referee, whom we appreciate for suggesting such an interesting notation. 968 SHOJI YOKURA

Here we consider the following commutative diagram:

(f 0)∗E / E

0 0 f X ×X V / V p0 s p z z X0 / XY f

0 0 (Note that the left diamond is a ﬁber square, thus f : X ×X V → V is 0 0 0 0 smooth and p : X ×X V → X is proper. Note that dim f = dim f and dim(s ◦ f 0) = dim s + dim f 0 = m + dim f.) (b) For a proper map g : Y 0 → Y , the (proper) pullback acting on the sec- s ∗ + ond factor V −→ Y , in an unusual way denoted by g : Mm,r(X,Y ) → 0 + Mm,r(X,Y ) , is deﬁned by

0 p s ∗ p◦g 0 s0 0 0 ∗ ([X ←− V −→ Y ; E]) g := [X ←−− V ×Y Y −→ Y ;(g ) E].

Here, we consider the following commutative diagram:

E o (g0)∗E

0 g 0 V o V ×Y Y s0 p s $ $ 0 XYYo g

0 0 0 (Note that the right diamond is a ﬁber square, thus s : V ×Y Y → Y 0 0 0 is smooth and g : V ×Y Y → V is proper, and dim s = dim s .)

Remark 4.12. (1) We emphasize that as to pushforward, proper pushforward is concerned with the first factor and smooth pushforward is concerned with the second factor, but as to pullback, the involving factors are exchanged. (2) We remark that when we deal with a smooth map f or g, both in pushforward and pullback, the first grading is added by the relative dimension dim f or dim g, but that when we deal with proper maps, the first grading is not changed. In both pushforward and pullback, the second grading (referring to the dimension of vector bundle) is not changed.

Proposition 4.13. The above three operations of product (•⊕ and •⊗), pushforward and pullback satisfy the following properties. COBORDISMBICYCLESOFVECTORBUNDLES 969

(A1) Product is associative: For three varieties X,Y,Z,W we have + (α •⊕ β) •⊕ γ = α •⊕ (β •⊕ γ) ∈ Mm+n+`,r+k+e(X,W ) , + (α •⊗ β) •⊗ γ = α •⊗ (β •⊗ γ) ∈ Mm+n+`,rke(X,W ) , + + + where α ∈ Mm,r(X,Y ) , β ∈ Mn,k(Y,Z) and β ∈ M`,e(Z,W ) , (A2) Pushforward is functorial : 0 0 00 (a) For two proper maps f1 : X → X , f2 : X → X , we have

(f2 ◦ f1)∗ = (f2)∗ ◦ (f1)∗ + 0 + where (f1)∗ : Mm,r(X,Y ) → Mm,r(X ,Y ) and 0 + 00 + (f2)∗ : Mm,r(X ,Y ) → Mm,r(X ,Y ) . 0 0 00 (b) For two smooth maps g1 : Y → Y , g2 : Y → Y we have

∗(g2 ◦ g1) = ∗(g1) ◦ ∗(g2) i.e., α ∗(g2 ◦ g1) = (α ∗(g1)) ∗(g2) 0 where ∗(g1): Mm,r(X,Y ) → Mm+dim g1,r(X,Y ) and 0 00 ∗(g2): Mm+dim g1,r(X,Y ) → Mm+dim g1+dim g2,r(X,Y ). (A2)’ Proper pushforward and smooth pushforward commute: For a proper map f : X → X0 and a smooth map g : Y → Y 0 we have 5 + ∗g ◦ f∗ = f∗ ◦ ∗g, i.e., (f∗α) ∗g = f∗(α ∗g) for α ∈ Mm,r(X,Y ) i.e, the following diagram commutes:

+ f∗ 0 + Mm,r(X,Y ) −−−−→ Mm,r(X ,Y )   ∗g ∗g y y 0 + 0 0 + Mm+dim g,r(X,Y ) −−−−→ Mm+dim g,r(X ,Y ) . f∗

(A3) Pullback is functorial: 0 0 00 (a) For two smooth maps f1 : X → X , f2 : X → X we have ∗ ∗ ∗ (f2 ◦ f1) = (f1) ◦ (f2) ∗ 00 + 0 + where (f2) : Mm,r(X ,Y ) → Mm+dim f2,r(X ,Y ) and ∗ 0 + + (f1) : Mm+dim f2,r(X ,Y ) → Mm+dim f2+dim f1,r(X,Y ) . 0 0 00 (b) For two proper maps g1 : Y → Y , g2 : Y → Y we have ∗ ∗ ∗ ∗ ∗ ∗ (g2 ◦ g1) = (g2) ◦ (g1) i.e., α (g2 ◦ g1) = (α (g2)) (g1) ∗ 0 + + where (g1): Mm,r(X,Y ) → Mm,r(X,Y ) and ∗ 00 + 0 + (g2): Mm,r(X,Y ) → Mm,r(X,Y ) . (A3)’ Proper pullback and smooth pullback commute: For a smooth map g : X0 → X and a proper map f : Y 0 → Y we have ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ + g ◦ f = f ◦ g , i.e., g (α f) = (g α) f for α ∈ Mm,r(X,Y )

5 The referee remarked that introducing the above unusual notation turns, e.g., the condition (A2)’ into a “bimodule-like” condition ((rm)s = r(ms) for an R-S-bimodule M with two rings R,S; r ∈ R, s ∈ M, m ∈ M) , which makes the proofs in the paper easier to follow. 970 SHOJI YOKURA

i.e, the following diagram commutes:

∗ + f 0 + Mm,r(X,Y ) −−−−→ Mm,r(X,Y )   ∗  ∗ g y yg 0 + 0 0 + Mm+dim g,r(X ,Y ) −−−−→ Mm+dim g,r(X ,Y ) . ∗f + (A12) Product and pushforward commute: Let α ∈ Mm,r(X,Y ) and β ∈ + Mn,k(Y,Z) . (a) For a proper morphism f : X → X0, 0 + f∗(α •⊕ β) = (f∗α) •⊕ β (∈ Mm+n,r+k(X ,Z) ), 0 + f∗(α •⊗ β) = (f∗α) •⊗ β (∈ Mm+n,rk(X ,Z) ), i.e., (for the sake of clarity), the following diagrams commute:

+ + •⊕ + Mm,r(X,Y ) ⊗ Mn,k(Y,Z) −−−−→ Mm+n,r+k(X,Z)   f∗×id +  f Mn,k(Y,Z) y y ∗ 0 + + 0 + Mm,r(X ,Y ) ⊗ Mn,k(Y,Z) −−−−→ Mm+n,r+k(X ,Z) . •⊕

+ + •⊗ + Mm,r(X,Y ) ⊗ Mn,k(Y,Z) −−−−→ Mm+n,rk(X,Z)   f∗×id +  f Mn,k(Y,Z) y y ∗ 0 + + 0 + Mm,r(X ,Y ) ⊗ Mn,k(Y,Z) −−−−→ Mm+n,rk(X ,Z) . •⊗ (b) For a smooth morphism g : Z → Z0, 0 + (α •⊕ β) ∗g = α •⊕ β ∗g (∈ Mm+n+dim g,r+k(X,Z ) ), 0 + (α •⊗ β) ∗g = α •⊗ β ∗g (∈ Mm+n+dim g,rk(X,Z ) ),

i.e., as to the product •⊕ (the case of •⊗ is omitted), the following diagram commutes:

+ + •⊕ + Mm,r(X,Y ) ⊗ Mn,k(Y,Z) −−−−→ Mm+n,r+k(X,Z)   id + ⊗ ∗g ∗g Mm,r(X,Y ) y y + 0 + 0 + Mm,r(X,Y ) ⊗ Mn+dim g,k(Y,Z ) −−−−→ Mm+n+dim g,r+k(X,Z ) . •⊕ + (A13) Product and pullback commute: Let α ∈ Mm,r(X,Y ) and + β ∈ Mn,k(Y,Z) . (a) For a smooth morphism f : X0 → X, ∗ ∗ 0 + f (α •⊕ β) = (f α) •⊕ β (∈ Mm+n+dim f,r+k(X ,Z) ), ∗ ∗ 0 + f (α •⊗ β) = (f α) •⊗ β (∈ Mm+n+dim f,rk(X ,Z) ), COBORDISMBICYCLESOFVECTORBUNDLES 971

i.e., as to the product •⊕, the following diagram commutes:

+ + •⊕ + Mm,r(X,Y ) ⊗ Mn,k(Y,Z) −−−−→ Mm+n,r+k(X,Z)

∗   f ×id +  f ∗ Mn,k(Y,Z) y y 0 + + 0 + Mm+dim f,r(X ,Y ) ⊗ Mn,k(Y,Z) −−−−→ Mm+n+dim,r+k(X ,Z) . •⊕

(b) For a proper morphism g : Z0 → Z,

∗ ∗ 0 + (α •⊕ β) g = α •⊕ β g (∈ Mm+n,r+k(X,Z ) ), ∗ ∗ 0 + (α •⊗ β) g = α •⊗ β g (∈ Mm+n,rk(X,Z ) ),

i.e., as to the product •⊕, the following diagram commutes:

+ + •⊕ + Mm,r(X,Y ) ⊗ Mn,k(Y,Z) −−−−→ Mm+n,r+k(X,Z)

∗   id + ⊗ g ∗ Mm,r(X,Y ) y y g + 0 + 0 + Mm,r(X,Y ) ⊗ Mn,k(Y,Z ) −−−−→ Mm+n,r+k(X,Z ) . •⊕

+ (A23) Pushforward and pullback commute: For α ∈ Mm,r(X,Y ) (a) (proper pushforward and proper pullback commute) For proper mor- 0 0 + phisms f : X → X and g : Y → Y and for α ∈ Mm,r(X,Y )

∗ ∗ 0 0 + (f∗α) g = f∗(α g)(∈ Mm,r(X ,Y ) ),

i.e., the following diagram commutes:

+ f∗ 0 + Mm,r(X,Y ) −−−−→ Mm,r(X ,Y )   ∗  ∗ gy y g 0 + 0 0 + Mm,r(X,Y ) −−−−→ Mm,r(X ,Y ) . f∗

(b) (smooth pushforward and smooth pullback commute) For smooth mor- 0 0 + phisms f : X → X and g : Y → Y and for α ∈ Mm,r(X,Y )

∗ ∗ 0 0 + f (α ∗g) = (f α) ∗g (∈ Mm+dim f+dim g,r(X ,Y ) ),

i.e., the following diagram commutes:

+ ∗g 0 + Mm,r(X,Y ) −−−−→ Mm+dim g,r(X,Y )   ∗  ∗ f y yf 0 + 0 0 + Mm+dim f,r(X ,Y ) −−−−→ Mm+dim f+dim g,r(X ,Y ) . ∗g 972 SHOJI YOKURA

fe Xe −−−−→ X00    g gey y X0 −−−−→ X f with f proper and g smooth, we have ∗ ∗ g f∗ = fe∗ge , i.e., the following diagram commutes:

0 + f∗ + Mm,r(X ,Y ) −−−−→ Mm,r(X,Y )   ∗  ∗ ge y yg + 00 + Mm+dim g,r(X,Ye ) −−−−→ Mm+dim g,r(X ,Y ) . fe∗ (Note that dim ge = dim g.) (d) (smooth pushforward and proper pullback “commute” in the following sense) For the following ﬁber square

fe Ye −−−−→ Y 00    g gey y Y 0 −−−−→ Y f 00 + with f proper and g smooth, for α ∈ Mm,r(X,Y ) we have ∗ ∗ (α ∗g) f = (α fe) ∗g,e i.e., the following diagram commutes:

00 + ∗g + Mm,r(X,Y ) −−−−→ Mm+dim g,r(X,Y )   ∗  ∗ fey y f + 0 + Mm,r(X, Ye) −−−−→ Mm+dim g,r(X,Y ) . ∗ge (Note that dim ge = dim g.) (A123) “Projection formula”: 0 + (a) For a smooth morphism g : Y → Y and α ∈ Mm,r(X,Y ) and 0 + β ∈ Mn,k(Y ,Z) , ∗ + (α ∗g) •⊕ β = α •⊕ g β (∈ Mm+n+dim g,r+k(X,Z) ), ∗ + (α ∗g) •⊗ β = α •⊗ g β (∈ Mm+n+dim g,rk(X,Z) ), COBORDISMBICYCLESOFVECTORBUNDLES 973

i.e., as to the product •⊕, the following diagram commutes:

∗ id + ×g + 0 + Mm,r (X,Y ) + + Mm,r(X,Y ) ⊗ Mn,k(Y ,Z) / Mm,r(X,Y ) ⊗ Mn+dim g,k(Y,Z)

∗g⊗id 0 + • Mn,k(Y ,Z) ⊕

0 + 0 + + Mm+dim g,r(X,Y ) ⊗ Mn,k(Y ,Z) / Mm+n+dim g,r+k(X,Z) . •⊕

0 + 0 + (b) For a proper map g : Y → Y , α ∈ Mm,r(X,Y ) and β ∈ Mn,k(Y ,Z) ,

∗ + (α g) •⊕ β = α •⊕ g∗β (∈ Mm+n,r+k(X,Z) ), ∗ + (α g) •⊗ β = α •⊗ g∗β (∈ Mm+n,rk(X,Z) ),

i.e., as to the product •⊕, the following diagram commutes:

id + ×g∗ + 0 + Mm,r(X,Y ) + + Mm,r(X,Y ) ⊗ Mn,k(Y ,Z) / Mm,r(X,Y ) ⊗ Mn,k(Y,Z)

∗ g⊗id 0 + • Mn,k(Y ,Z) ⊕ 0 + 0 + + Mm,r(X,Y ) ⊗ Mn,k(Y ,Z) / Mm+n,r+k(X,Z) . •⊕

Proof. It is straightforward. For the sake of readers’ convenience, we give proofs to (c) and (d) of A23 and the “projection formula” with respect to the product •⊕. 0 p s 0 + (c) Let [X ←− V −→ Y ; E] ∈ Mm,r(X ,Y ) and consider the following commutative diagrams:

(g0)∗E

# Xe ×X0 V E p0 g0

{ # Xe V fe ge p s ~ $ z X00 X0 Y f g ! X z 974 SHOJI YOKURA

0 0 ∗ 0 p s p s◦g 0 ∗ fe∗ge ([X ←− V −→ Y ; E]) = fe∗([Xe ←− Xe ×X0 V −−→ Y ;(g ) E] 0 0 00 fe◦p s◦g 0 ∗ = [X ←−− Xe ×X0 V −−→ Z;(g ) E] f◦p = g∗([X ←−− V −→s Z; E]) ∗ 0 p s = g (f∗([X ←− V −→ Y ; E]) ∗ 0 p s = g f∗([X ←− V −→ Y ; E]).

∗ ∗ Hence, we have that g f∗ = fe∗ge .

p s 00 00 + (d) Let [X ←− V −→ Y ; E] ∈ Mm,r(X,Y ) and consider the following commutative diagrams:

(f 0)∗E

{ E V ×Y 00 Ye f 0 s0

{ # V Ye p s fe ge $ z XY 00 Y 0

g f ~ $ Y

p s 00 ∗ p g◦s ∗ ([X ←− V −→ Y ; E]∗g) f = [X ←− V −−→ Y ; E] f 0 0 p◦f ge◦s 0 0 ∗ = [X ←−− V ×Y 00 Ye −−→ Y ;(f ) E] 0 p◦f s0 0 ∗ = [X ←−− V ×Y 00 Ye −→ Ye;(f ) E]∗ge p s 00 ∗ = ([X ←− V −→ Y ; E] fe)∗g.e

∗ ∗ Hence, we have that (α ∗g) f = (α fe) ∗g.e

p s 0 q t (A123): Let α = [X ←− V −→ Y ; E], β = [Y ←− W −→ Z; F ].

(a) For the ﬁrst projection formula, we consider the following commutative diagram. COBORDISMBICYCLESOFVECTORBUNDLES 975

(q00)∗E ⊕ (g0 ◦ s0)∗F

0 0 ∗ E V ×Y (Y ×Y 0 W ) (g ) F q00 s0

w ( $ V Y ×Y 0 W F p s q0 g0 XY' u $ W q t g

) Y 0 z Z p g◦s 0 Since α ∗g = [X ←− V −−→ Y ; E], we have

0 0 0 p◦q t◦(g ◦s ) 00 ∗ 0 0 ∗ α ∗g •⊕ β = [X ←−− V ×Y 0 W −−−−−→ Z;(q ) E ⊕ (g ◦ s ) F ] 0 0 0 p◦q 0 (t◦g )◦s 00 ∗ 0 ∗ 0 ∗ = [X ←−− V ×Y (Y ×Y 0 W ) −−−−−→ Z;(q ) E ⊕ (s ) ((g ) F )] 0 0 p s q 0 t◦g 0 ∗ = [X ←− V −→ Y ; E] •⊕ [Y ←− Y ×Y 0 W −−→ Z;(g ) F ] ∗ = α •⊕ g β. (b) For the second projection formula, we consider the following commutative diagram: (q0)∗(g0)∗E ⊕ (s00)∗F

0 ∗ 0 (g ) E (V ×Y Y ) ×Y 0 W F q0 s00 v { 0 ' E V ×Y Y W g0 s0 q t V z ) Y 0 v Z p s g ~ XY$ u

0 0 ∗ p◦g 0 s 0 0 ∗ Then, since α g = [X ←−− V ×Y Y −→ Y ;(g ) E], we have

0 0 ∗ p◦g ◦q 0 t◦s00 0 ∗ 0 ∗ 00 ∗ α g •⊕ β = [X ←−−−− (V ×Y Y ) ×Y 0 W −−→ Z;(q ) (g ) E ⊕ (s ) F ] 976 SHOJI YOKURA

0 0 p◦(g ◦q ) t◦s00 0 0 ∗ 00 ∗ = [X ←−−−−− V ×Y W −−→ Z;(g ◦ q ) E ⊕ (s ) F ] p Y g◦q t = [X ←− V −→; E] •⊕ [Y ←−− W −→ Z; F ]

= α •⊕ g∗β.

idX idX + Remark 4.14. 1X = [X ←−− X −−→ X] ∈ M0,0(X,X) satisﬁes that 1X •⊕ + α = α for any element α ∈ Mm,r(X,Y ) and β •⊕ 1X = β for any element + idX idX β ∈ Mm,r(Y,X) . As to the product •⊗, let 11X = [X ←−− X −−→ X; OX ] ∈ + M0,0(X,X) (where OX is the trivial line bundle) satisﬁes that 11X •⊗ α = α + for any element α ∈ Mm,r(X,Y ) and β •⊗ 11X = β for any element β ∈ + Mm,r(Y,X) .

+ Remark 4.15. Mm,r(X,Y ) with the product •⊕ considered shall be denoted + + by Mm,r(X,Y )⊕ and Mm,r(X,Y ) with the product •⊗ shall be denoted by + Mm,r(X,Y )⊗.

prop f Remark 4.16. In a way similar to that of constructing OMsm (X −→ Y ) generated h by the isomorphism classes [V −→ X; L1, ··· ,Lr] such that h is proper and f ◦ h h is smooth and Li’s are line bundles over V , we can replace [V −→ X; L1, ··· ,Lr] h by the isomorphism classes [V −→ X; E] with line bundles {L1, ··· ,Lr} being replaced by one vector bundle E. Then in [4] (cf.[2,3]) T. Annala and the author have generalized Lee–Pandharipande’s algebraic cobordism ω∗,∗(X) of vector bundles [18](also see [21]) to a bivariant-theoretic analogue Ω∗,∗(X → Y ) in a way similar to that of Annala’s construction of the bivariant derived algebraic cobordism Ω∗(X → Y ).

5. Cobordism bicycles of ﬁnite tuples of line bundles Here we consider a “bi-variant” analogue of Levine–Morel’s construction of algebraic cobordism Ω∗(X) [20] via cobordism bicycles of ﬁnite tuples of line bundles. In [20], Levine and Morel consider what they call a cobordism cycle

p [M −→ X; L1, ··· Lr],

p which is the isomorphism class of (M −→ X; L1, ··· Lr) where M is smooth and p irreducible, p : M → X is proper and Li’s are line bundles over M. (M −→ 0 0 p 0 0 X; L1, ··· Lr) and (M −→ X; L1, ··· Lr) are called isomorphic if there exists an COBORDISMBICYCLESOFVECTORBUNDLES 977 isomorphism h : M → M 0 such that (1) the following diagram commutes M p

! h =∼ X =

0 p M 0 ∼ ∼ and (2) there exists a bijection σ : {1, 2, ··· , r} = {1, 2, ··· , r} such that Li = ∗ h Lσ(i). So, here we consider a “bicycle” version of this cobordism cycle. Namely we consider the isomorphism class of a cobordism bicycle of a ﬁnite tuple of line bundles, instead of one vector bundle.

0 0 p s p 0 s 0 0 Definition 5.1. (X ←− V −→ Y ; L1, ··· ,Lr) and (X ←− V −→ Y ; L1, ··· ,Lr) are called isomorphic if the following conditions hold: p s p0 (1) There exists an isomorphism h : V =∼ V 0 such (X ←− V −→ Y ) =∼ (X ←− s0 V 0 −→ Y ) as correspondences (as in (1) of Definition 4.3), ∼ ∼ (2) There exists a bijection σ : {1, 2, ··· , r} = {1, 2, ··· , r} such that Li = ∗ h Lσ(i). p s The isomorphism class of (X ←− V −→ Y ; L1, ··· ,Lr) is called simply a cobordism bicycle (instead of a cobordism bicycle of line bundles) and denoted by p s [X ←− V −→ Y ; L1, ··· ,Lr]. p s It is clear that when Y = pt is a point the cobordism bicycle [X ←− V −→ p pt; L1, ··· ,Lr] is the same as the cobordism cycle [V −→ X; L1, ··· ,Lr]. Definition 5.2. We define Z i(X,Y ) to be the free abelian group generated by the set of isomorphism classes of cobordism bicycle p s [X ←− V −→ Y ; L1,L2, ··· ,Lr], such that −i+r = dim s = dim V −dim Y , modulo the following additive relation

p1 s1 p2 s2 0 0 [X ←− V1 −→ Y ; L1, ··· ,Lr] + [X ←− V2 −→ Y ; L1, ··· ,Lr] p1tp2 s1ts2 0 0 := [X ←−−− V2 −−−→ Y ; L1 t L1, ··· ,Lr t Lr]. Remark 5.3. Such a grading is due to the requirement that for Y = pt we want i to have Z (X, pt) = Z−i(X). Here we note that V is smooth, since s : V → pt is smooth. According to the deﬁnition ([20, Deﬁnition 2.1.6]) of grading of Levine-Morel’s algebraic pre-cobordism Z∗(X), the degree (or dimension) of the p cobordism cycle [V −→ X; L1, ··· ,Lr] ∈ Z∗(X) is dim V − r, i.e. 978 SHOJI YOKURA

p [V −→ X; L1, ··· ,Lr] ∈ Z−i(X) ⇐⇒ −i = dim V − r, namely, −i + r = dim V . The following definitions are similar to those in the case of cobordism bicycles of vector bundles, but we write them down for the sake of convenience. Definition 5.4. (1) (Product of cobordism bicycles) We define the product of cobordism bicycles as follows: • : Z i(X,Y ) ⊗ Z j(Y,Z) → Z i+j(X,Z)

p1 s1 p2 s2 [X ←− V1 −→ Y ; L1,L2, ··· ,Lr] • [Y ←− V2 −→ Z; M1,M2, ··· ,Mk] p1 s1 p2 s2 ∗ ∗ ∗ ∗ := [(X ←− V1 −→ Y ) ◦ (Y ←− V2 −→ Z); pe2 L1, ··· , pe2 Lr, se1 M1, ··· se1 Mk].

V1 ×Y V2 . pe2 se1 z $ V1 V2 p1 s1 p2 s2 ~ XYZ$ z (2) (Proper pushforward and smooth pushforward of cobordism bicycles) (a) For a proper map f : X → X0, the (proper) pushforward (with respect i i 0 to the ﬁrst factor) f∗ : Z (X,Y ) → Z (X ,Y ) is deﬁned by

p s 0 f◦p s f∗([X ←− V −→ Y ; L1,L2, ··· ,Lr]) := [X ←−− V −→ Y ; L1,L2, ··· ,Lr]. (b) For a smooth map g : Y → Y 0, the (smooth) pushforward (with i i+dim g 0 respect to the second factor) ∗g : Z (X,Y ) → Z (X,Y ) is defined by p s p g◦s 0 ([X ←− V −→ Y ; L1,L2, ··· ,Lr]) ∗g := [X ←− V −−→ Y ; L1,L2, ··· ,Lr]. (3) (Smooth pullback and proper pullback of cobordism bicycles) (a) For a smooth map f : X0 → X, the (smooth) pullback (with respect to the first factor) f ∗ : Z i(X,Y ) → Z i+dim f (X0,Y ) is defined by ∗ p s f ([X ←− V −→ Y ; L1,L2, ··· ,Lr]) 0 0 0 p 0 s◦f 0 ∗ 0 ∗ 0 ∗ := [X ←− X ×X V −−→ Y ;(f ) L1, (f ) L2, ··· , (f ) Lr]. Here we consider the following commutative diagram:

0 0 f X ×X V / V p0 s p z z X0 / XY f COBORDISMBICYCLESOFVECTORBUNDLES 979

(b) For a proper map g : Y 0 → Y , the (proper) pullback (with respect to the second factor) ∗g : Z i(X,Y ) → Z i(X,Y 0) is deﬁned by p s ∗ ([X ←− V −→ Y ; L1,L2, ··· ,Lr]) g 0 p◦g 0 s0 0 0 ∗ 0 ∗ 0 ∗ := [X ←−− V ×Y Y −→ Y ;(g ) L1, (g ) L2, ··· , (g ) Lr]. Here we consider the following commutative diagram:

0 g 0 V o V ×Y Y s0 p s $ $ 0 XYYo g Remark 5.5. As in the case of cobordism bicycles of vector bundles, the involve- ment of factors are exchanged in pushforward and pullback for proper morphisms and smooth morphisms, and also, in both pushforward and pullback, as long as smooth morphisms are involved, the grading is added by the relative dimension of the smooth morphism. Remark 5.6. In the case of cobordism bicycles of vector bundles, we have two kind of products by Whitney sum and tensor product. However, in the present case of cobordism bicycles, the product is a kind of “Whitney sum”, or a “mock” Whitney sum, i.e., the sum of two ﬁnite sets of line bundles. Clearly we have the following proposition, as in the case of cobordism bicycles of vector bundles in §4. Proposition 5.7. The above three operations satisfy the following properties. i (A1) Product is associative: For three varieties X,Y,Z,W and α ∈ Z (X,Y ), β ∈ Z j(Y,Z), γ ∈ Z k(Z,W ), we have (α • β) • γ = α • (β • γ)(∈ Z i+j+k(X,W ))

(A2) Pushforward is functorial : 0 0 00 (a) For two proper maps f1 : X → X , f2 : X → X and i i 0 (f1)∗ : Z (X,Y ) → Z (X ,Y ), i 0 i 00 (f2)∗ : Z (X ,Y ) → Z (X ,Y ), we have

(f2 ◦ f1)∗ = (f2)∗ ◦ (f1)∗. 0 0 00 (b) For two smooth maps g1 : Y → Y , g2 : Y → Y and i i+dim g 0 ∗(g1): Z (X,Y ) → Z 1 (X,Y ), i+dim g 0 i+dim g +dim g 00 ∗(g2): Z 1 (X,Y ) → Z 1 2 (X,Y ), we have

∗(g2 ◦ g1) = ∗(g1) ◦ ∗(g2), i.e., α ∗(g2 ◦ g1) = (α ∗(g1)) ∗(g2).

(A2)’ Proper pushforward and smooth pushforward commute: For a proper map f : X → X0 and a smooth map g : Y → Y 0 we have

∗g ◦ f∗ = f∗ ◦ ∗g, i.e., (f∗α) ∗g = f∗(α ∗g), simply denoted by f∗α ∗g, 980 SHOJI YOKURA

i.e, the following diagram commutes:

f Z i(X,Y ) −−−−→∗ Z i(X0,Y )   ∗g ∗g y y Z i+dim g(X,Y 0) −−−−→ Z i+dim g(X0,Y 0). f∗

(A3) Pullback is functorial: 0 0 00 (a) For two smooth maps f1 : X → X , f2 : X → X and ∗ i+dim f 0 i+dim f +dim f (f1) : Z 2 (X ,Y ) → Z 2 1 (X,Y ), ∗ i 00 i+dim f 0 (f2) : Z (X ,Y ) → Z 2 (X ,Y ), we have ∗ ∗ ∗ (f2 ◦ f1) = (f1) ◦ (f2) . 0 0 00 (b) For two proper maps g1 : Y → Y , g2 : Y → Y and ∗ i 0 i (g1): Z (X,Y ) → Z (X,Y ), ∗ i 00 i 0 (g2): Z (X,Y ) → Z (X,Y ), we have ∗ ∗ ∗ ∗ ∗ (g2 ◦ g1) = (g2) ◦ (g1), i.e., α (g2 ◦ g1) = (α (g2)) ∗(g1).

(A3)’ Proper pullback and smooth pullback commute: For a smooth map g : X0 → X and a proper map f : Y 0 → Y we have g∗ ◦ ∗f = ∗f ◦ g∗, i.e., g∗(α ∗f) = (g∗α) ∗f, simply denoted by g∗α ∗f, i.e, the following diagram commutes: i.e, the following diagram commutes:

∗f Z i(X,Y ) −−−−→ Z i(X,Y 0)   ∗  ∗ g y yg Z i+dim g(X0,Y ) −−−−→ Z i+dim g(X0,Y 0). ∗f i (A12) Product and pushforward commute: For α ∈ Z (X,Y ) and β ∈ Z j(Y,Z). (a) For a proper morphism f : X → X0, i+j 0 f∗(α • β) = (f∗α) • β (∈ Z (X ,Z)), i.e., (for the sake of clarity we write down) the following diagram commutes: • Z i(X,Y ) ⊗ Z j(Y,Z) −−−−→ Z i+j(X,Z)   f∗⊗id j   Z (Y,Z)y yf∗ Z i(X0,Y ) ⊗ Z j(Y,Z) −−−−→ Z i+j(X0,Z). • (b) For a smooth morphism g : Z → Z0, i+j+dim g 0 (α • β) ∗g = α • β ∗g (∈ Z (X,Z )), COBORDISMBICYCLESOFVECTORBUNDLES 981

i.e., the following diagram commutes: • Z i(X,Y ) ⊗ Z j(Y,Z) −−−−→ Z i+j(X,Z)   id i ⊗ ∗g ∗g Z (X,Y ) y y Z i(X,Y ) ⊗ Z j+dim g(Y,Z0) −−−−→ Z i+j+dim g(X,Z0). • i j (A13) Product and pullback commute: For α ∈ Z (X,Y ) and β ∈ Z (Y,Z). (a) For a smooth morphism f : X0 → X,

f ∗(α • β) = (f ∗α) • β (∈ Z i+j+dim f (X0,Z)), i.e., the following diagram commutes: • Z i(X,Y ) ⊗ Z j(Y,Z) −−−−→ Z i+j(X,Z)

∗   f ⊗id j   ∗ Z (Y,Z)y yf Z i+dim f (X0,Y ) ⊗ Z j(Y,Z) −−−−→ Z i+j+dim f (X0,Z). • (b) For a proper morphism g : Z0 → Z,

(α • β) ∗g = α • β ∗g (∈ Z i+j(X,Z0)), i.e., the following diagram commutes: • Z i(X,Y ) ⊗ Z j(Y,Z) −−−−→ Z i+j(X,Z)

∗   id i ⊗ g ∗ Z (X,Y ) y y g Z i(X,Y ) ⊗ Z j(Y,Z0) −−−−→ Z i+j(X,Z0). • i (A23) Pushforward and pullback commute: For α ∈ Z (X,Y ) (a) (proper pushforward and proper pullback commute) For proper morphisms f : X → X0 and g : Y 0 → Y ,

∗ ∗ i 0 0 ∗ (f∗α) g = f∗(α g)(∈ Z (X ,Y )), simply denoted by f∗α g, i.e., the following diagram commutes:

f Z i(X,Y ) −−−−→∗ Z (X0,Y )   ∗  ∗ gy y g Z i(X,Y 0) −−−−→ Z i(X0,Y 0). f∗ (b) (smooth pushforward and smooth pullback commute)For smooth morphisms f : X0 → X and g : Y → Y 0,

∗ ∗ i+dim f+dim g 0 0 ∗ f (α ∗g) = (f α) ∗g (∈ Z (X ,Y )), simply denoted by f α ∗g, 982 SHOJI YOKURA

i.e., the following diagram commutes: g Z i(X,Y ) −−−−→∗ Z i+dim g(X,Y 0)   ∗  ∗ f y yf Z i+dim f (X0,Y ) −−−−→ Z i+dim f+dim g(X0,Y 0). ∗g (c) (proper pushforward and smooth pullback “commute” in the following sense) For the following ﬁber square

fe Xe −−−−→ X00    g gey y X0 −−−−→ X f with f proper and g smooth, we have ∗ ∗ g f∗ = fe∗ge , i.e., the following diagram commutes: f Z i(X0,Y ) −−−−→∗ Z i(X,Y )   ∗  ∗ ge y yg Z i+dim g(X,Ye ) −−−−→ Z i+dim g(X00,Y ). fe∗ (Note that dim ge = dim g.) (d) (smooth pushforward and proper pullback “commute” in the following sense) For the following ﬁber square

fe Ye −−−−→ Y 00    g gey y Y 0 −−−−→ Y f with f proper and g smooth, we have ∗ ∗ (α ∗g) f = (α fe) ∗g,e i.e., the following diagram commutes: g Z i(X,Y 00) −−−−→∗ Z i+dim(X,Y )   ∗  ∗ fey y f Z i(X, Ye) −−−−→ Z i+dim g(X,Y 0). ∗ge (Note that dim ge = dim g.) (A123) “Projection formula”: COBORDISMBICYCLESOFVECTORBUNDLES 983

(a) For a smooth morphism g : Y → Y 0 and α ∈ Z i(X,Y ) and β ∈ Z j(Y 0,Z),

∗ i+j+dim g (α ∗g) • β = α • g β (∈ Z (X,Z)), i.e., the following diagram commutes:

id ⊗g∗ Z i(X,Y ) Z i(X,Y )+ ⊗ Z j(Y 0,Z) −−−−−−−−−→ Z i(X,Y ) ⊗ Z j+dim g(Y,Z)   ∗g⊗id j  • Z (Y,Z)y y Z i+dim g(X,Y 0) ⊗ Z j(Y 0,Z) −−−−→ Z i+j+dim g(X,Z). •

(b) For a proper map g : Y 0 → Y , α ∈ Z i(X,Y ) and β ∈ Z j(Y 0,Z),

∗ i+j (α g) • β = α • g∗β (∈ Z (X,Z)), the following diagram commutes:

id ⊗g Z i(X,Y ) ∗ Z i(X,Y ) ⊗ Z j(Y 0,Z) −−−−−−−−−→ Z i(X,Y ) ⊗ Z j(Y,Z)

∗   g⊗id j 0  • Z (Y ,Z)y y Z i(X,Y 0) ⊗ Z j(Y 0,Z) −−−−→ Z i+j(X,Z). •

idX idX 0 Remark 5.8. 11X := [X ←−− X −−→ X] ∈ Z (X,X) satisﬁes that 11X • α = α ∗ ∗ for any element α ∈ Z (X,Y ) and β • 11X = β for any element β ∈ Z (Y,X). The following fact is emphasized for a later use.

Lemma 5.9. (Pushforward-Product Property for Units (abbr. PPPU)) For the following ﬁber square

V ×Y W p e se

V z $ W

s p $ Y z with s : V → Y smooth and p : W → Y proper, we have

(11 s) • p 11 = (p 11 ) s = p (11 s) ∈ (V,W ). V ∗ ∗ W e∗ V ×Y W ∗e e∗ V ×Y W ∗e Z Here we have the homomorphisms

∗ ∗ ∗ ∗ ∗s : Z (V,V ) → Z (V,Y ), p∗ : Z (W, W ) → Z (Y,W ) 984 SHOJI YOKURA and the following commutative diagram

∗ Z (V ×Y W, V ×Y W ) pe∗ ∗se

∗ t ∗* Z (V,V ×Y W ) Z (V ×Y W, W )

s p ∗e * t e∗ Z ∗(V,W ). In the same way as in Levine–Morel’s algebraic cobordism, we define Chern operators as follows: Definition 5.10. For a line bundle L over X or a line bundle M over Y , we first define the following Chern classes:

idX idX 1 c1(L) := [X ←−− X −−→ X; L] ∈ Z (X,X),

idY idY 1 c1(M) := [Y ←−− Y −−→ Y ; M] ∈ Z (Y,Y ). Then the “Chern class operators” i i+1 i i+1 c1(L)• : Z (X,Y ) → Z (X,Y ), • c1(M): Z (X,Y ) → Z (X,Y ) are respectively deﬁned by p s p s ∗ c1(L)•([X ←− V −→ Y ; L1,L2, ··· ,Lr]) = [X ←− V −→ Y ; L1,L2, ··· ,Lr, p L],

p s p s ∗ ([X ←− V −→ Y ; L1,L2, ··· ,Lr])•c1(M) = [X ←− V −→ Y ; L1,L2, ··· ,Lr, s M]. Lemma 5.11. The above Chern class operators satisfy the following properties. (1) (identity): If L and L0 are line bundles over X and isomorphic and if M and M 0 are line bundles over Y and isomorphic, then we have 0 i i+1 c1(L)• = c1(L )• : Z (X,Y ) → Z (X,Y ), 0 i i+1 • c1(M) = • c1(M ): Z (X,Y ) → Z (X,Y ). (2) (commutativity): If L and L0 are line bundles over X and if M and M 0 are line bundles over Y , then we have 0 0 i i+2 c1(L) • c1(L )• = c1(L ) • c1(L)• : Z (X,Y ) → Z (X,Y ), 0 0 i i+2 •c1(M) • c1(M ) = •c1(M ) • c1(M): Z (X,Y ) → Z (X,Y ). (3) (compatibility with product) Let L be a line bundle over X and N be a line bundle over Z. For α ∈ Z i(X,Y ) and β ∈ Z j(Y,Z), we have c1(L) • (α • β) = c1(L) • α • β, (α • β) • c1(N) = α • β • c1(N) . COBORDISMBICYCLESOFVECTORBUNDLES 985

(4) (compatibility with pushforward = “projection formula”) (which is “similar” to [11, Theorem 3.2, (c) (Projection formula)]): For a proper map f : X → X0 and a line bundle L over X0 and for a smooth map g : Y → Y 0 and a line bundle M over Y 0 we have that for α ∈ Z i(X,Y ) ∗ ∗ f∗ c1(f L) • α) = c1(L) • f∗α, α • c1(g M) ∗g = α ∗g • c1(M). (5) (compatibility with pullback =“pullback formula”)(which is “similar” to [11, Theorem 3.2, (d) (Pull-back)]): For a smooth map f : X0 → X and a line bundle L over X and for a proper map g : Y 0 → Y and a line bundle M over Y we have that for α ∈ Z i(X,Y ) ∗ ∗ ∗ ∗ ∗ ∗ f c1(L) • α = c1(f L) • f α, α • c1(M) g = α g • c1(g M). Proof. We show only (4) and (5). First we observe that as to (4) ∗ ∗ ∗ ∗ f∗c1(f L) = c1(L) f, c1(g M) ∗g = g c1(M), (5.12) and as to (5) ∗ ∗ ∗ ∗ c1(f L) ∗f = f c1(L), c1(g M) ∗g = c1(M) g. (5.13) Indeed, the ﬁrst one of (5.12) can be seen as follows:

∗ idX idX ∗ f∗c1(f L) = f∗([X ←−− X −−→ X; f L]) f id = [X0 ←− X −−→X X; f ∗L]

id 0 id 0 = [X0 ←−−−X X0 −−−→X X0; L] ∗f (by the deﬁnition of ∗f) ∗ = c1(L) f. Similarly, we can see the other three equalities which are left to the reader. As to (4), we show the ﬁrst one: ∗ ∗ f∗ c1(f L) • α) = f∗c1(f L) • α (by A12 (a) ) ∗ = c1(L) f • α (by (5.12)

= c1(L) • f∗α. (by A123 (Projection formula) (a) ) As to (5), we show the second one: ∗ ∗ α • c1(M) g = α • c1(M) g (by A13 (b) ) ∗ = α • c1(g M) ∗g (by (5.13) ∗ ∗ = α g • c1(g M). (by A123 (Projection formula) (b)) As to the compatibility with pullback, we observe the following fact concerning the unit: Lemma 5.14 (Pullback Property for Unit (abbr. PPU)). For a smooth map f : X0 → X and a line bundle L over X and for a proper map g : Y 0 → Y and a line i i bundle M over Y we have that for 11X ∈ Z (X,X) and 11Y ∈ Z (Y,Y ) ∗ ∗ ∗ i+dim f+1 0 c1(f L) • (f 11X ) = (f 11X ) • c1(L) ∈ Z (X ,X), 986 SHOJI YOKURA

∗ ∗ ∗ i+1 0 (11Y g) • c1(g M) = c1(M) • (11Y g) ∈ Z (Y,Y ). Proof. We prove only the first equality, i.e., the case of smooth maps, since the idX idX second equality can be proved in the same way. Since 11X = [X ←−− X −−→ X], ∗ it follows from the definition of the smooth pullback f 11X (see Definition 5.4) ∗ 0 idX0 0 f that we have f 11X = [X ←−−− X −→ X]:

f X0 / X idX0 idX id } ~ X X0 / XX f Hence, we have

∗ 0 idX0 0 f 0 idX0 0 f ∗ (f 11X ) • c1(L) = [X ←−−− X −→ X] • c1(L) = [X ←−−− X −→ X; f L]. On the other hand, since f ∗L is a line bundle over X0, clearly we have

∗ ∗ ∗ 0 idX0 0 f c1(f L) • (f 11X ) = c1(f L) • [X ←−−− X −→ X]

0 idX0 0 f ∗ ∗ = [X ←−−− X −→ X; (idX0 ) f L]

id 0 f = [X0 ←−−−X X0 −→ X; f ∗L]. ∗ ∗ ∗ Thus, we obtain c1(f L) • (f 11X ) = (f 11X ) • c1(L). 0 If we let f : X → X be the identity idX : X → X, we get the following: Corollary 5.15 (Commutativity of the unit and Chern class). Let L be a line bundle over X. Then we have

c1(L) • 11X = 11X • c1(L).

Remark 5.16. The following observations plays key roles later. Let L1,L2, ··· Lr be line bundles over V . Then it follows from Lemma 5.11 (3) that ! c1(L1)• c1(L2)• ···• c1(Lr)•α ··· = c1(L1)•c1(L2)•··· c1(Lr) •α, which is simply denoted by c1(L1) • c1(L2) •··· c1(Lr) • α. Then it follows from the deﬁnitions that we have

idV idV [V ←−− V −−→ V ; L1,L2, ··· ,Lr] = c1(L1) • c1(L2) •···• c1(Lr) • 11V ,

idV idV [V ←−− V −−→ V ; L1,L2, ··· ,Lr] = 11V • c1(L1) • c1(L2) •···• c1(Lr).

In fact, it follows from Corollary 5.15 that 11V can be placed at any place;

idV idV [V ←−− V −−→ V ; L1,L2, ··· ,Lr]

= c1(L1) •···• c1(Lj) • 11V • c1(Lj+1) •···• c1(Lr). COBORDISMBICYCLESOFVECTORBUNDLES 987

Hence, we have p s [X ←− V −→ Y ; L1,L2, ··· ,Lr] = p∗ c1(L1) • c1(L2) •···• c1(Lr) • 11V ∗s,

p s [X ←− V −→ Y ; L1,L2, ··· ,Lr] = p∗ 11V • c1(L1) • (L2) •···• c1(Lr) ∗s, and in general, p s [X ←− V −→ Y ; L1,L2, ··· ,Lr] (5.17) = p∗ c1(L1) •···• c1(Lj) • 11V • c1(Lj+1) •···• c1(Lr) ∗s.

Deﬁnition 5.18 (Bi-variant theory). An association B assigning to a pair (X,Y ) a graded abelian group B∗(X,Y ) is called a bi-variant theory provided that (1) it is equipped with the following three operations (1) (Product) • : Bi(X,Y ) × Bj(Y,Z) → Bi+j(X,Z) (2) (Pushforward) 0 i i 0 (a) For a proper map f : X → X , f∗ : B (X,Y ) → B (X ,Y ). 0 i i+dim g 0 (b) For a smooth map g : Y → Y , ∗g : B (X,Y ) → B (X,Y ). (3) (Pullback) (a) For a smooth map f : X0 → X, f ∗ : Bi(X,Y ) → Bi+dim f (X0,Y ). (b) For a proper map g : Y 0 → Y , ∗g : Bi(X,Y ) → Bi(X,Y 0). (2) the three operations satisfy the following nine properties as in Proposition 5.7:

(A1) Product is associative. (A2) Pushforward is functorial. ((a), (b)) (A2)’ Proper pushforward and smooth pushforward commute. (A3) Pullback is functorial. ((a), (b)) (A3)’ Proper pullback and smooth pullback commute. (A12) Product and pushforward commute. ((a), (b)) (A13) Product and pullback commute. ((a), (b)) (A23) Pushforward and pullback commute. ((a), (b), (c), (d)) (A123) Projection formula. ((a), (b)) 0 (3) B has units, i.e., there is an element 1X ∈ B (X,X) such that 1X • α = α for any element α ∈ B(X,Y ) and β • 1X = β for any element β ∈ B(Y,X). (4) B satisﬁes PPPU and PPU. (5) B is equipped with the Chern class operators satisfying the properties in Lemma 5.11. Deﬁnition 5.19. Let B, B0 be two bi-variant theories on a category V .A Grothen- dieck transformation from B to B0, γ : B → B0 is a collection of homomorphisms B(X,Y ) → B0(X,Y ) for a pair (X,Y ) in the category V , which preserves the above three basic operations and the Chern class operator:

(1) γ(α •B β) = γ(α) •B0 γ(β), (2) γ(f∗α) = f∗γ(α) and γ(α ∗g) = γ(α) ∗g, (3) γ(g∗α) = g∗γ(α) and γ(α ∗f) = γ(α) ∗f, (4) γ(c1(L) • α) = c1(L) •B γ(α) and γ(α • c1(M)) = γ(α) •B c1(M). 988 SHOJI YOKURA

Theorem 5.20. The above Z ∗(−, −) is the universal one among bi-variant theories in the following sense. Given any bi-variant theory B∗(−, −), there exists a unique Grothendieck transformation

∗ ∗ γB : Z (−, −) → B (−, −) such that γB(11V ) = 1V ∈ B(V,V ) for any variety V .

Proof. Let B be a bi-variant theory. From now on we just simply write γ for γB ∗ and • for •B unless some possible confusion with those for the theory Z . Then, using the observation (5.17) made in Remark 5.16, we deﬁne

γ : Z i(X,Y ) → Bi(X,Y )

p s i by, for [X ←− V −→ Y ; L1, ··· ,Lr] ∈ Z (X,Y ), p s γ([X ←− V −→ Y ; L1, ··· ,Lr]) (5.21) = γ p∗ c1(L1) •···• c1(Lj) • 11V • c1(Lj+1) •···• c1(Lr) ∗s := p∗ c1(L1) •···• c1(Lj) • 1V • c1(Lj+1) •···• c1(Lr) ∗s.

We show that this transformation satisﬁes the above four properties: (1) γ(α • β) = γ(α) • γ(β): Let p1 s1 i α = [X ←− V1 −→ Y ; L1, ··· ,Lr] ∈ Z (X,Y ),

p2 s2 j β = [Y ←− V2 −→ Z; M1, ··· ,Mk] ∈ Z (Y,Z). Then, by the deﬁnition, we have

p1 s1 p2 s2 [X ←− V1 −→ Y ; L1,L2, ··· ,Lr] • [Y ←− V2 −→ Z; M1,M2, ··· ,Mk] p1 s1 p2 s2 ∗ ∗ ∗ ∗ := [(X ←− V1 −→ Y ) ◦ (Y ←− V2 −→ Z); pe2 L1, ··· , pe2 Lr, se1 M1, ··· se1 Mk] p1◦pe2 s2◦se1 ∗ ∗ ∗ ∗ = [X ←−−− V1 ×Y V2 −−−→ Z; pe2 L1, ··· , pe2 Lr, se1 M1, ··· se1 Mk].

V1 ×Y V2 pe2 se1 z $ V1 V2 p1 s1 p2 s2 ~ XYZ$ z Hence, we have γ(α • β)

p1◦pe2 s2◦se1 ∗ ∗ ∗ ∗ = γ([X ←−−− V1 ×Y V2 −−−→ Z; pe2 L1, ··· , pe2 Lr, se1 M1, ··· se1 Mk]) COBORDISMBICYCLESOFVECTORBUNDLES 989 = γ (p ◦ p ) c (p ∗L ) •··· c (p ∗L ) • 11 1 e2 ∗ 1 e2 1 1 e2 r V1×Y V2 ∗ ∗ • c1(se1 M1) ···• c1(se1 Mk) ∗(s2 ◦ se1)

(5.21) = (p ◦ p ) c (p ∗L ) •··· c (p ∗L ) • 1 1 e2 ∗ 1 e2 1 1 e2 r V1×Y V2 ∗ ∗ • c1(se1 M1) ···• c1(se1 Mk) ∗(s2 ◦ se1) (A ) =2 (p ) (p ) c (p ∗L ) •··· c (p ∗L ) • 1 1 ∗ e2 ∗ 1 e2 1 1 e2 r V1×Y V2 ∗ ∗ • c1(se1 M1) ···• c1(se1 Mk) ∗(se1)∗(s2) (A )0 =2 (p ) (p ) c (p ∗L ) •··· c (p ∗L ) • 1 1 ∗ e2 ∗ 1 e2 1 1 e2 r V1×Y V2 ∗ ∗ • c1(se1 M1) ···• c1(se1 Mk) ∗(se1) ∗(s2). Then, by applying the property (4) of Lemma 5.11 successively with respect to line bundles L1, ··· ,Lr and M1, ··· ,Mk, we get (p ) c (p ∗L ) •··· c (p ∗L ) • 1 • c (s ∗M ) ···• c (s ∗M ) (s ) e2 ∗ 1 e2 1 1 e2 r V1×Y V2 1 e1 1 1 e1 k ∗ e1 = c (L ) •··· c (L ) • (p ) 1 (s ) • c (M ) ···• c (M ). 1 1 1 r e2 ∗ V1×Y V2 ∗ e1 1 1 1 k Here it follows from Lemma 5.9 (PPPU) that we have (p ) 1 (s ) = 1 (s ) • (p ) 1 . e2 ∗ V1×Y V2 ∗ e1 V1 ∗ 1 2 ∗ V2 Hence, the above equalities continue as follows: = (p1)∗ c1(L1) ··· c1(Lr) • 1V1 ∗(s1) • (p2)∗1V2 • c1(M1) ··· c1(Mk) ∗(s2)

0 (A2) = (p1)∗ c1(L1) ··· c1(Lr) • 1V1 ∗(s1) • (p2)∗1V2 • c1(M1) ··· c1(Mk) ∗(s2) (A12)(a) = (p1)∗ c1(L1) ··· c1(Lr) • 1V1 ∗(s1) • (p2)∗1V2 • c1(M1) ··· c1(Mk) ∗(s2)

(A12)(b) = (p1)∗ c1(L1) ··· c1(Lr) • 1V1 ∗(s1) • (p2)∗1V2 • c1(M1) ··· c1(Mk) ∗(s2)

(A12) = (p1)∗ c1(L1) ··· c1(Lr) • 1V1 ∗(s1) • (p2)∗ 1V2 • c1(M1) ··· c1(Mk) ∗(s2) 990 SHOJI YOKURA

p1 s1 p2 s2 = γ([X ←− V1 −→ Y ; L1, ··· ,Lr]) • γ([Y ←− V2 −→ Z; M1, ··· ,Mk]) = γ(α) • γ(β). Hence, we have γ(α • β) = γ(α) • γ(β).

(2) γ(f∗α) = f∗γ(α) and γ(α ∗f) = γ(α) ∗f: p s i Let α = [X ←− V −→ Y ; L1,L2, ··· ,Lr] ∈ Z (X,Y ). 0 (i) γ(f∗α) = f∗γ(α): Let f : X → X be a proper map. Then we have

0 f◦p s γ(f∗α) = γ([X ←−− V −→ Y ; L1, ··· ,Lr]) = γ (f ◦ p)∗ c1(L1) •··· c1(Lr) • 11V ∗s

(5.21) = (f ◦ p)∗ c1(L1) •··· c1(Lr) • 1V ∗s

(A2)(a) = f∗p∗ c1(L1) •··· c1(Lr) • 1V ∗s ! 0 (A2) = f∗ p∗ c1(L1) •··· c1(Lr) • 1V ∗s

p s = f∗γ([X ←− V −→ Y ; L1,L2, ··· ,Lr])

= f∗γ(α). 0 (ii) γ(α ∗f) = γ(α) ∗f: Let f : Y → Y be a smooth map. Then we have

p f◦s 0 γ(α ∗f) = γ([X ←− V −−→ Y ; L1,L2, ··· ,Lr]) = γ p∗ c1(L1) •··· c1(Lr) • 11V ∗(f ◦ s)

(5.21) = p∗ c1(L1) •··· c1(Lr) • 1V ∗(f ◦ s)

(A2)(b) = p∗ c1(L1) •··· c1(Lr) • 1V ∗s ∗f

0 (A2) = p∗ c1(L1) •··· c1(Lr) • 1V ∗s ∗f p s = γ([X ←− V −→ Y ; L1,L2, ··· ,Lr]) ∗f

= γ(α) ∗f. (3) γ(f ∗α) = f ∗γ(α) and γ(α ∗f) = γ(α) ∗f: p s i Let α = [X ←− V −→ Y ; L1,L2, ··· ,Lr] ∈ Z (X,Y ). (i) γ(f ∗α) = f ∗γ(α): Let f : X0 → X be a smooth map and consider the following diagram:

0 0 f X ×X V / V p0 s p z z X0 / XY f COBORDISMBICYCLESOFVECTORBUNDLES 991

0 0 ∗ 0 p 0 s◦f 0 ∗ 0 ∗ 0 ∗ γ(f α) = γ([X ←− X ×X V −−→ Y ;(f ) L1, (f ) L2, ··· , (f ) Lr]) 0 0 ∗ 0 ∗ 0 0 = γ (p )∗ 11X ×X V • c1 (f ) L1 ···• c1 (f ) Lr ∗(s ◦ f ) (5.21) 0 0 ∗ 0 ∗ 0 0 = (p )∗ 1X ×X V • c1 (f ) L1 ···• c1 (f ) Lr ∗(s ◦ f ) (A2)(b) 0 0 ∗ 0 ∗ 0 0 = (p )∗ 1X ×X V • c1 (f ) L1 ···• c1 (f ) Lr ∗(f ) ∗s ! 0 0 ∗ 0 ∗ 0 0 = (p )∗ 1X ×X V • c1 (f ) L1 ···• c1 (f ) Lr ∗(f ) ∗s.

By applying the property (4) of Lemma 5.11 successively with respect to line bundles L1, L2, ··· ,Lr, the above equalities continue as follows: 0 0 0 = (p )∗ 1X ×X V ∗(f ) • c1(L1) ···• c1(Lr) ∗s

(A12)(a) 0 0 0 = (p )∗1X ×X V ∗(f ) • c1(L1) ···• c1(Lr) ∗s. Here we note that 0 0 0 0 (p )∗1X ×X V ∗(f ) = 1X ∗f • p∗1V (by Lemma 5.9 (PPPU)) ∗ = 1X0 • f p∗1V (by A123 (a)). Thus the above equalities continue as follows:

∗ = 1X0 • f p∗1V • c1(L1) ···• c1(Lr) ∗s

∗ = f p∗1V • c1(L1) ···• c1(Lr) ∗s (since 1X0 is the unit)

(A13)(a) ∗ = f p∗1V • c1(L1) ···• c1(Lr) ∗s

(A23)(b) ∗ = f p∗1V • c1(L1) ···• c1(Lr) ∗s

(A12)(a) ∗ = f p∗(1V • c1(L1) ···• c1(Lr)) ∗s

(A23)(b) ∗ = f p∗(1V • c1(L1) ···• c1(Lr)) ∗s

∗ p s = f γ([X ←− V −→ Y ; L1,L2, ··· ,Lr]) = f ∗γ(α). Here we note that in fact we can show the above “indirectly”, using Lemma 5.14 (PPU) and Lemma 5.11 (5) as well:

∗ = f p∗1V • c1(L1) • c1(L2) •···• c1(Lr) ∗s

(A23)(c) 0 0 ∗ = (p )∗(f ) 1V • c1(L1) • c1(L2) •···• c1(Lr) ∗s 992 SHOJI YOKURA

0 (A2) 0 0 ∗ = (p )∗ (f ) 1V • c1(L1) •c1(L2) •···• c1(Lr) ∗s | {z } 0 0 ∗ 0 ∗ = (p )∗ c1((f ) L1) • (f ) 1V • c1(L2) •···• c1(Lr) ∗s | {z } (by applying (PPU) for L1) 0 0 ∗ 0 ∗ 0 ∗ = (p )∗ c1((f ) L1) • c1((f ) L2) • (f ) 1V • c1(L3) •···• c1(Lr) ∗s | {z } (by applying (PPU) for L2) 0 0 ∗ 0 ∗ 0 ∗ 0 ∗ = (p )∗ c1((f ) L1) • c1((f ) L2) •···• c1((f ) Lr) • (f ) 1V ∗s | {z } (by applying (PPU) for the rest) 0 0 ∗ 0 ∗ 0 ∗ = (p )∗ c1((f ) L1) • c1((f ) L2) •···• (f ) (c1(Lr) • 1V ) ∗s

(by applying Lemma 5.11 (5) for Lr) 0 0 ∗ = (p )∗ (f ) c1(L1) •···• c1(Lr) • 1V ) ∗s (by applying Lemma 5.11 (5) successively) 0 (A2) 0 0 ∗ = (p )∗(f ) c1(L1) •···• c1(Lr) • 1V ) ∗s

(A23)(c) ∗ = f p∗ c1(L1) •···• c1(Lr) • 1V ∗s

(A23)(b) ∗ = f p∗ c1(L1) •···• c1(Lr) • 1V ∗s

∗ p s = f γ([X ←− V −→ Y ; L1,L2, ··· ,Lr]) = f ∗γ(α).

(ii) γ(α ∗f) = γ(α) ∗f: Let f : Y 0 → Y be a proper map. The proof is similar to the above. For the sake of the reader’s convenience, we write down the proof. Let f : Y 0 → Y be a proper map and consider the following diagram:

0 f 0 V o V ×Y Y s0 p s $ $ XYYo 0 f

0 ∗ p◦f 0 s0 0 0 ∗ 0 ∗ 0 ∗ γ(α f) = γ([X ←−− X ×X V −→ Y ;(f ) L1, (f ) L2, ··· , (f ) Lr]) ! 0 0 ∗ 0 ∗ 0 0 = (p ◦ f )∗ c1 (f ) L1 •··· c1 (f ) Lr • 1V ×Y Y ) ∗(s ) ! (A2) 0 0 ∗ 0 ∗ 0 0 = p∗(f )∗ c1 (f ) L1 •··· c1 (f ) Lr • 1V ×Y Y ) ∗(s ). COBORDISMBICYCLESOFVECTORBUNDLES 993

By applying the property (4) of Lemma 5.11 successively with respect to line bundles L1, L2, ··· ,Lr, the above equalities continue as follows: 0 0 0 = p∗ c1(L1) •··· c1(Lr) • (f )∗1V ×Y Y ∗(s )

(A12) 0 0 0 = p∗ c1(L1) •··· c1(Lr) • (f )∗1V ×Y Y ∗(s ) . Here it follows from Lemma 5.9 (PPPU)) that we have 0 0 0 (f )∗1V ×Y Y = 1V ∗s • f∗1Y . Thus, the above equalities continue as follows: = p∗ c1(L1) •··· c1(Lr) • (1V ∗s • f∗1Y 0 )

(A123)(b) ∗ = p∗ c1(L1) •··· c1(Lr) • (1V ∗s f • 1Y 0 )

∗ = p∗ c1(L1) •··· c1(Lr) • 1V ∗s f (since 1Y 0 is the unit)

(A13) ∗ = p∗ c1(L1) •··· c1(Lr) • 1V ∗s f

0 (A2) ∗ = p∗ c1(L1) •··· c1(Lr) • 1V ∗s f

∗ = p∗ c1(L1) •··· c1(Lr) • 1V ∗s f = γ(α) ∗f. As done for another proof of (i) using using Lemma 5.14 (PPU) and Lemma 5.11 (5), we can show the above in a similar way. For the sake of the reader we write down the proof. ∗ = p∗ c1(L1) •···• c1(Lr−1) • c1(Lr) • 1V ∗s f

(A23)(d) ∗ 0 0 = p∗ c1(L1) •···• c1(Lr−1) • c1(Lr) • 1V (s ) ∗(f )

0 (A2) ∗ 0 0 = p∗ c1(L1) •···• c1(Lr−1) • c1(Lr) • 1V (s ) ∗(f ) | {z } ∗ 0 0 ∗ 0 = p∗ c1(L1) •···• c1(Lr−1) • 1V (s ) •c1((s ) Lr) ∗(f ) | {z } (by applying (PPU) for Lr) ∗ 0 0 ∗ 0 ∗ 0 = p∗ c1(L1) •···• 1V (s ) • c1((s ) Lr−1) • c1((s ) Lr) ∗(f )

(by applying (PPU) for Lr−1) ∗ 0 0 ∗ 0 ∗ 0 = p∗ 1V (s ) • c1((s ) L1) •···• c1((s ) Lr) ∗(f ) | {z } (by applying (PPU) for the rest) ∗ 0 0 ∗ 0 = p∗ 1V • c1(L1) (s ) •···• c1((s ) Lr) ∗(f )

(by applying Lemma 5.11 (5) for L1) 994 SHOJI YOKURA

∗ 0 0 = p∗ 1V • c1(L1) •···• c1(Lr) (s ) ∗(f ) (by applying Lemma 5.11 (5) successively) 0 (A2) ∗ 0 0 = p∗ 1V • c1(L1) •···• c1(Lr) (s ) ∗(f )

(A23)(d) ∗ = p∗ 1V • c1(L1) •···• c1(Lr) ∗s f

(A23)(a) ∗ = p∗ 1V • c1(L1) •··· c1(Lr) ∗s f = γ(α) ∗f.

(4) γ(c1(L) • α) = c1(L) • γ(α) and γ(α • c1(M)) = γ(α) • c1(M): Let α = p s i [X ←− V −→ Y ; L1,L2, ··· ,Lr] ∈ Z (X,Y ). (i) γ(c1(L) • α) = c1(L) • γ(α): Let L be a line bundle over X. Then we have p s ∗ γ(c1(L) • α) = γ([X ←− V −→ Y ; L1, ··· ,Lr, p L]) ∗ = p∗ c1(L1) •··· c1(Lr) • c1(p L) • 1V ∗s

(Lemma 5.11 (2)) ∗ = p∗ c1(p L) • c1(L1) •··· c1(Lr) • 1V ∗s ∗ = p∗ c1(p L) • c1(L1) •··· c1(Lr) • 1V ∗s

(Lemma 5.11 (4)) = c1(L) • p∗ c1(L1) •··· c1(Lr) • 1V ∗s = c1(L) • p∗ c1(L1) •··· c1(Lr) • 1V ∗s

= c1(L) • γ(α).

(ii) γ(α • c1(M)) = γ(α) • c1(M): Let M be a line bundle over Y . Then we have p s ∗ γ(α • c1(M)) = γ([X ←− V −→ Y ; L1,L2, ··· ,Lr, s M]) ∗ = p∗ c1(L1) ···• c1(Lr) • 1V • c1(s M) ∗s (see Rmeark 5.16)

0 (A2) ∗ = p∗ c1(L1) ···• c1(Lr) • 1V • c1(s M) ∗s

(Lemma 5.11 (4)) = p∗ c1(L1) ···• c1(Lr) • 1V ∗s • c1(M) (A12) = p∗ c1(L1) ···• c1(Lr) • 1V ∗s • c1(M)

= γ(α) • c1(M). (5) The uniqueness of the Grothendieck transformation γ : Z → B follows from the compatibility of pushforward and the Chern class operator and the requirement that the unit is mapped to the unit. Indeed, let γ0 : Z → B be a such a COBORDISMBICYCLESOFVECTORBUNDLES 995

Grothendieck transformation. Then we have 0 0 p s γ (α) = γ ([X ←− V −→ Y ; L1,L2, ··· ,Lr]) ! 0 = γ p∗ c1(L1) •··· c1(Lr) • 11V ∗s

0 = p∗ c1(L1) •··· c1(Lr) • γ (11V ) ∗s = p∗ c1(L1) •··· c1(Lr) • 1V ) ∗s

= γB(α). Remark 5.22. The “forget” map deﬁned in (1.2) gives rise to the following canonical homomorphism

pro f ∗ f : OMsm (X −→ Y ) → Z (X,Y ) deﬁned by

p p f◦p f([V −→ Y ; L1, ··· ,Lr]) := [X ←− V −−→ Y ; L1, ··· ,Lr]. Then it follows from the deﬁnitions that the following diagrams are commutative: (1) As to the product •:

prop f prop g • prop g◦f OMsm (X −→ Y ) ⊗ OMsm (Y −→ Z) −−−−→ OMsm (X −−→ Z)     f⊗fy yf Z i(X,Y ) ⊗ Z j(Y,Z) −−−−→ Z i+j(X,Z). • (2) As to the pushforward:

prop g◦f f∗ prop g OMsm (X −−→ Z) −−−−→ OMsm (Y −→ Z)     fy yf Z ∗(X,Z) −−−−→ Z ∗(Y,Z). f∗ (3) As to the Chern class operator:

prop f Φ(L) prop f OMsm (X −→ Y ) −−−−→ OMsm (X −→ Y )     fy yf Z ∗(X,Y ) −−−−→ Z ∗(X,Y ). c1(L)• As to the pullback we cannot expect a canonical commutative diagram. Indeed we consider the following ﬁber square 996 SHOJI YOKURA

g00 V 0 −−−−→ V   0  h y yh g0 X0 −−−−→ X   0  f y yf Y 0 −−−−→ Y. g Then we have the following diagram, which does not necessarily commute:

∗ 0 prop f g prop 0 f 0 OMsm (X −→ Y ) / OMsm (X −→ Y ) (5.23)

f f Z ∗(X,Y ) / Z ∗(X0,Y 0). (g0)∗◦( ∗g)

∗ h (f ◦ g )([V −→ X;L1, ··· ,Lr]) 0 0 h 0 00 ∗ 00 ∗ = f([V −→ X ;(g ) L1, ··· , (g ) Lr]) 0 0 0 0 h 0 f ◦h 0 00 ∗ 00 ∗ = [X ←− V −−−→ Y ;(g ) L1, ··· , (g ) Lr]).

0 ∗ ∗ h (g ) ◦ ( g) ◦ f ([V −→ X; L1, ··· ,Lr])

0 ∗ ∗ h f◦h = (g ) ◦ ( g) ([X ←− V −−→ Y ; L1, ··· ,Lr]) 0 ∗ h f◦h ∗ = (g ) ([X ←− V −−→ Y ; L1, ··· ,Lr]) g

0 00 0 0 00 0 h ◦gf 0 0 f ◦h ◦gf 0 00 ∗ 00 ∗ 00 ∗ 00 ∗ = [X ←−−− V ×V V −−−−−→ Y ;(ge ) (g ) L1, ··· , (ge ) (g ) Lr]). Here we consider the following ﬁber square:

00 0 0 gf 0 V ×V V −−−−→ V   00  00 gfy yg V 0 −−−−→ V. g00 Thus, in general, we have that f ◦ g∗ 6= (g0)∗ ◦ ( ∗g) ◦ f in the above diagram (5.23). For the sake of convenience, we list the properties for Z ∗(X, Y) with Y fixed and Z ∗(X,Y ) with X fixed. Note that to emphasize the target Y or the source X being fixed we denote them in bold, Y and X, respectively. Proposition 5.24. (1) Z ∗(X, Y): COBORDISMBICYCLESOFVECTORBUNDLES 997

(a) (Proper pushforward is covariantly functorial): For two proper maps 0 0 00 i i 0 f1 : X → X , f2 : X → X , (f1)∗ : Z (X, Y) → Z (X , Y), we i 0 i 00 have (f2)∗ : Z (X , Y) → Z (X , Y) and

(f2 ◦ f1)∗ = (f2)∗ ◦ (f1)∗. (b) (Smooth pullback is contravariantly functorial): For two smooth maps 0 0 00 ∗ i+dim f 0 f1 : X → X , f2 : X → X , we have (f1) : Z 2 (X , Y) → i+dim f +dim f ∗ i 00 i+dim f 0 Z 2 1 (X, Y), (f2) : Z (X , Y) → Z 2 (X , Y) and ∗ ∗ ∗ (f2 ◦ f1) = (f1) ◦ (f2) . (c) For a line bundle L over X we have the Chern operator: i i+1 c1(L)• : Z (X, Y) → Z (X, Y). 0 0 Moreover, if L and L are isomorphic, then c1(L)• = c1(L ) • . (d) (Proper pushforward and smooth pullback commute)

fe Xe −−−−→ X00    g gey y X0 −−−−→ X f with f proper and g smooth, we have ∗ ∗ g f∗ = fe∗ge , i.e., the following diagram commutes: f Z i(X0, Y) −−−−→∗ Z i(X, Y)   ∗  ∗ ge y yg Z i+dim g(X,e Y) −−−−→ Z i+dim g(X00, Y). fe∗ (Note that dim ge = dim g.) (e) (compatibility with proper pushforward (“projection formula”)): For a proper map f : X → X0 and a line bundle L over X0, we have that for α ∈ Z i(X, Y) ∗ f∗ c1(f L) • α = c1(L) • f∗α. (f) (compatibility with smooth pullback (“pullback formula”)): For a smooth map f : X0 → X and a line bundle L over X, we have that for α ∈ Z i(X, Y) ∗ ∗ ∗ f c1(L) • α = c1(f L) • f α. (g) (commutativity): If L and L0 are line bundles over X, then we have 0 0 i i+2 c1(L) • c1(L )• = c1(L ) • c1(L)• : Z (X, Y) → Z (X, Y), 998 SHOJI YOKURA

(2) Z ∗(X,Y ): (a) (Smooth pushforward is covariantly functorial): For two smooth maps 0 0 00 g1 : Y → Y , g2 : Y → Y , we have i i+dim g 0 ∗(g1): Z (X,Y ) → Z 1 (X,Y ), i+dim g 0 i+dim g +dim g 00 ∗(g2): Z 1 (X,Y ) → Z 1 2 (X,Y ) and

∗(g2 ◦ g1) = ∗(g1) ◦ ∗(g2). (b) (Proper pullback is contravariantly functorial): For two proper maps 0 0 00 ∗ i 0 i g1 : Y → Y , g2 : Y → Y and (g1): Z (X,Y ) → Z (X,Y ) ∗ i 00 i 0 and (g2): Z (X,Y ) → Z (X,Y ) ∗ ∗ ∗ (g2 ◦ g1) = (g2) ◦ (g1). (c) For a line bundle M over Y we have the Chern operator: i i+1 •c1(M): Z (X,Y ) → Z (X,Y ). 0 0 Moreover, if M and M are isomorphic, then •c1(M) = •c1(L ). (d) (Smooth pushforward and proper pullback commute)

fe Ye −−−−→ Y 00    g gey y Y 0 −−−−→ Y f with f proper and g smooth, we have ∗ ∗ f ∗g = ∗ge f,e i.e., the following diagram commutes: g Z i(X,Y 00) −−−−→∗ Z i+dim(X,Y )   ∗  ∗ fey y f Z i(X, Ye) −−−−→ Z i+dim g(X,Y 0). ∗ge (Note that dim ge = dim g.) (e) (compatibility with smooth pushforward (“projection formula”)): For a smooth map g : Y → Y 0 and a line bundle M over Y 0 we have that for α ∈ Z i(X,Y )

∗ α • c1(g M) ∗g = α ∗g • c1(M). (f) (compatibility with proper pullback (“pullback formula”)): For a proper map g : Y 0 → Y and a line bundle M over Y we have that for α ∈ Z i(X,Y )

∗ ∗ ∗ α • c1(M) g = α g • c1(g M). COBORDISMBICYCLESOFVECTORBUNDLES 999

(g) (commutativity): If M and M 0 are line bundles over Y , then we have

0 0 i i+2 •c1(M) • c1(M ) = •c1(M ) • c1(M): Z (X,Y ) → Z (X,Y ),

Acknowledgements: The author would like to thank the anonymous referee for his/her careful reading of the manuscript, valuable comments, and construc- tive suggestions. This work was supported by JSPS KAKENHI Grant Numbers JP16H03936 and JP19K03468.

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(Shoji Yokura) GRADUATE SCHOOLOF SCIENCEAND ENGINEERING,KAGOSHIMA UNIVER- SITY, 21-35 KORIMOTO 1-CHOME,KAGOSHIMA 890-0065, JAPAN [email protected] This paper is available via http://nyjm.albany.edu/j/2020/26-41.html.