A Universal Bivariant Theory and Cobordism Groups · A UNIVERSAL BIVARIANT THEORY AND COBORDISM GROUPS SHOJI YOKURA(∗) Dedicated to Professor Mutsuo Oka on the occasion of his 60th

ESI The Erwin Schrodinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, Austria

A Universal Bivariant Theory and Cobordism Groups

Shoji Yokura

Vienna, Preprint ESI 2066 (2008) November 14, 2008

Supported by the Austrian Federal Ministry of Education, Science and CultureAvailable via http://www.esi.ac.at

A UNIVERSAL BIVARIANT THEORY AND COBORDISM GROUPS

SHOJI YOKURA(∗)

Dedicated to Professor Mutsuo Oka on the occasion of his 60th birthday

ABSTRACT. This is a survey on a universal bivariant theoryMC

S(X → Y ), which is a

prototype of a bivariant analogue of Levine–Morel’s algebraic cobordism, and its applica-tion to constructing a bivariant theoryFΩ(X → Y ) of cobordism groups. Before givingsuch a survey, we recall the genus such as signature, which isthe main important invariantdefined on the cobordism group, i.e, a ring homomorphism fromthe cobordism group to acommutative ring with a unit. We capture the Euler–Poincar´e characteristic and genera asa drastic generalization of the very naturalcounting finites sets.

1. INTRODUCTION

The (oriented) cobordism groupΩ∗ was introduced by Rene Thom [Th] in 1950’s andit was extended by Michael Atiyah [At] to the (oriented) cobordism theoryMSO∗(X) ofa topological spaceX, which is a generalized cohomology theory. It is defined by ThomspectraMSO(n). As a covariant or homology-like version ofMSO∗(X), Atiyah [At]introduced the bordism theoryMSO∗(X) geometrically in a quite simple manner, com-pared with the “spectral” definition of the cobordism theory. If we replaceSO(n) by theother groupsO(n), U(n), Spin(n), we get the corresponding cobordism and bordism the-ories.

The cobordism theoryMSO∗(X) is not “geometrically defined” unlike singular coho-mology, de Rham cohomology,K-theory, although the bordism theoryMSO∗(X) is verymuch geometrically simply defined. This kind of “drawback” pops up crucially when onedeals with the so-called elliptic cohomology theory. Note that algebraic topologists havebeen looking for a more geometrically described definition of the elliptic cohomology. Fora very recent survey on the elliptic cohomology, see Jacob Lurie’s survey paper [Lu], andalso see other papers and/or books [Hat, La2, Mi-Ra, Se, Ti].

Daniel Quillen introduced the notion of(complex ) oriented cohomology theoryon thecategory of differential manifolds [Qui] and this notion can be formally extended to thecategory of smooth schemes in algebraic geometry. VladimirVoevodsky has introducedalgebraic cobordism (now calledhigher algebraic cobordism), which was used in his proofof Milnor’s conjecture [Voe]. Marc Levine and Fabien Morel constructedthe universal oneof oriented cohomology theories, which they also callalgebraic cobordism, and have inves-tigated furthermore (see [Le1, Le2, LM1, LM2, LM3] and also see [Mer] for a condensedreview). Levine–Morel’s algebraic cobordism isthe universal oriented Borel–Moore func-tor satisfying some geometric axioms.

Mathematics Subject Classification 2000: 55N35, 55N22, 14C17, 14C40, 14F99, 19E99.Key words and phrases: bivaraint theory, bordism, cobordism, Borel–Moore functor, Euler–Poincare charac-

teristic, genus.(*) Partially supported by Grant-in-Aid for Scientific Research (No. 19540094), the Ministry of Education,

Culture, Sports, Science and Technology (MEXT), Japan.

1

2 SHOJI YOKURA(∗)

William Fulton and Robert MacPherson have introduced the notion of bivariant theoryas acategorical framework for the study of singular spaces, which is the title of their AMSMemoir book [FM] (see also Fulton’s book [Fu]). A bivariant theory is a unification ofcovariant functor such as a homology theory and a contravariant functor such as a coho-mology theory. A typical example is Fulton–MacPherson’s bivariant homology theoryH,whose associated contravariant functorH∗ is the usual singular cohomology theory andwhose associated covariant functorH∗ is the Borel–Moore homology theory,not the usualsingular homology theory. The main objective of [FM] is bivariant-theoretic Riemann–Roch’s or bivariant analogues of various theorems of Grothendieck–Riemann–Roch type.

We have recently constructed (prototypes of) bivariant-theoretic analogues of the abovetwo cobordism groups ([Yo1, Yo3])D There are two naıve motivations for this work:

(1) Levine–Morel’s algebraic cobordism is a covariant theory, thus we want to knowits contravariant version so that it is unified into its bivariant one.

(2) The definition of the cobordism theoryMSO∗(X) is not simple as that of the bor-dism theoryMSO∗(X), thus we want another contravariant version ofMSO∗(X)so that it is also unified into a bivariant-theoretic analogue of MSO∗(X). (Itwould be hopefully related to the problem of a geometric description of the ellip-tic cohomology.)

A key ingredient is what we calla universal bivariant theory, which is the universal oneamong the bivariant theories with a nice canonical orientation, or what could be calleda“motivic” bivariant theory.

In this paper we make a quick survey on these bivariant analogues after reviewing theEuler number and the genus, which is in a sense a drastic generalization of the Euler num-ber.

2. EULER–POINCARE CHARACTERISTIC AND GENERA

The genus is a homomorphism from the cobordism ring to another ring; thus these twonotions are two aspects of one thing in a sense. So, before going into the main parts ofthe paper in the next sections, in this section we recall or review how natural the notion ofEuler number or Euler–Poincare characteristic is and we see that the notion of genus is a“drastic” generalization of the Euler–Poincare characteristic, although the Euler–Poincarecharacteristic isnot a genus. Also we see that theordinary (co)homologyor more cor-rectly the Borel–Moore homology theoryconstructed from the ordinary (co)homology the-ory is in a sense for the Euler–Poincare characteristic andextraordinary or generalized(co)homology theoriesare for genera, in particular the most interesting one is theellipticgenus and the elliptic cohomology.

The cardinalityc(S), a nonnegative integer, of a finite setS satisfies the followingproperties:

(1) A ∼= B (set-isomorphism) impliesc(A) = c(B),(2) c(A ⊔ B) = c(A) + c(B),(3) c(A × B) = c(A) · c(B),(4) c(pt) = 1. (Herept denotes one point space.)

Or one could say that the isomorphism classes[S] of finite sets satisfies the above prop-erties except the property (4). Setting[φ] := 0 and [pt] := 1 gives us the nonnegativeintegers.

Now, let us consider the following similar problem for topological spaces:

A UNIVERSAL BIVARIANT THEORY AND COBORDISM GROUPS 3

Problem 2.1. Can one define a cardinalityχtop(W ) of a topological spaceW such thatthe above property (1) is replaced by the following “topological” one and the rest of theproperties are unchanged ? Namely, does there existχtop with the following properties ontopological spaces ?:

(1) A ∼= B (homeomorphism) impliesχtop(A) = χtop(B),(2) χtop(A ⊔ B) = χtop(A) + χtop(B),(3) χtop(A × B) = χtop(A) · χtop(B),(4) χtop(pt) = 1.

Since considering the cardinalityc(S) of a finite setS is the same as considering thecardinalityχtop(S) of the finite setS with the discrete topology, we just use the symbolχtop even for finite sets.

Remark 2.2. In the case of finite sets (with discrete topology), the property (2) can bewithout any caution replaced by the usual one:“inclusion-exclusion formula”

χtop(A ∪ B) = χtop(A) + χtop(B) − χtop(A ∩ B).

However, in the general case of topological spaces one has tobe a bit careful. In theproperty (2)A⊔B does not necessarilymean a direct sum of two topological spacesA andB, or the disjoint union of two topological spaces. One shouldnote that if it were so, theproperty (2) in general could not be replaced by the above “inclusion-exclusion formula.”It really means that if a topological space(X, T ) is decomposed intoX = A⊔B, then forthe topological subspaces(A, T |A) and(B, T |B), whereTS is the relative topology of asubsetS ⊂ X, we have the equality:

χtop((A ⊔B, T )) = χtop((A, T |A)) + χtop((B, T |B)).

This of course already takes care of even the case whenA ⊔B is the disjoint union of twotopological spaces. Thus, the above “inclusion-exclusionformula” means that ifA ∪ B isa topological space with a topologyT , then

χtop((A ∪ B, T )) = χtop((A, T |A)) + χtop((B, T |B)) − χtop((A ∩ B, T |A∩B)).

In this sense, even in the case of topological spaces the property (2) can be replaced bythe above “inclusion-exclusion formula.” However, we stick to the above simple “disjointunion” formula (2) for the sake of what follows later, i.e.,genera.

Now, we consider the looked-for cardinalityχtop for the 1-dimensional Euclidean spaceR1 and apply the property (2) to the decomposition ofR1:

R1 = (−∞, 0)⊔ 0 ⊔ (0,∞).

Which implies that

χtop(R1) = χtop((−∞, 0)) + χtop(0) + χtop((0,∞)).

Hence we have

−χtop(0) = χtop((−∞, 0)) + χtop((0,∞)) − χtop(R1).

SinceR ∼= (−∞, 0) ∼= (0,∞), it follows from (1) and (4) that

χtop(R1) = −χtop(0) = −1.

Hence, it follows from (3) that

χtop(Rn) = (−1)n.

So, in particular, ifX is a finiteCW -complex, i.e., a compact Hausdorff spaceX witha cellular structure, by applying the property (2) to the decomposition ofX into all opencells, we get

χtop(X) =∑

n

(−1)n♯n(X)

4 SHOJI YOKURA(∗)

where♯n(X) denotes the number ofn-dimensional open cells, since eachn-dimensionalopen cell is homeomorphic toRn. Namely,χtop(X) is nothing but the so-called Euler–Poincare characteristic of theCW -complexX. Of course we need to show that such acardinality exists, in particular it is independent of the decomposition of a given topologicalspace, and we know that the existence of such a cardinality isguaranteed at least by the(co)homology theory, to be more precisely, theordinary (co)homology theory. Here wejust recall an ordinary cohomology theory, since the homology version is a covariant one.

An ordinary cohomology theory is a sequenceHnn∈Z of contravariant functorsHn

from the category of pairs(X, A) of topological spaces to the category of abelian groups,together with a natural transformation∂ : Hn(A) → Hn+1(X, A) for eachn. These arerequired to satisfy the following axioms:

Homotopy Axiom : If f, g : (X, A) → (Y, B) are homotopic, then for eachn f∗ = g∗ :Hn(Y, B) → Hn(X, A).

Excision Axiom : If (X, A) is a pair andU ⊂ X such thatU is contained in the interiorA of A, then the inclusion mapj : (X − U, A− U) → (X, A) induces an isomorphism ;

j∗ : Hn(X, A) ∼= Hn(X − U, A − U).

Exactness Axiom: For any pair(X, A) with inclusion mapsi : A ⊂ X andj : X ⊂(X, A) there is a long exact sequence:

· · · → Hi(X, A) → Hn(X) → Hn(A) → Hn+1(X, A) → · · · .

Dimension Axiom: Let pt be the one-point space; thenHn(pt) = 0 for all n 6= 0.

H0(pt) is called the coefficient group of the given ordinary cohomology theoryH∗.

For a given ordinary cohomology theoryH∗, the H∗- Euler–Poincare characteristicχH∗(X) of a topological spaceX is defined to be

χH∗(X) :=∑

n

(−1)n dimR Hn(X) ⊗ R

provided that it is well-defined. ThisH∗- Euler–Poincare characteristicχH∗satisfies thefollowing properties for the category of topological spaces:

(1) A ∼= B (homeomorphism) impliesχH∗(A) = χH∗(B),(2) If (A ⊔B; A, B) is proper, i.e.,H∗(A) ∼= H∗(A ⊔B, B) andH∗(B) ∼= H∗(A ⊔

B, A), thenχH∗(A ⊔ B) = χH∗(A) + χH∗(B),(3) χH∗(A × B) = χH∗(A) · χH∗(B),(4) χH∗(pt) = 1.

The “inclusion-exclusion formula” version of (2) is the following: If (A ∪ B; A, B) isproper, i.e.,H∗(A, A ∩ B) ∼= H∗(A ∪ B, B) andH∗(B, A ∩ B) ∼= H∗(A ∪ B, A), thenχH∗(A∪B) = χH∗(A)+χH∗(B)−χH∗(A∩B). Which follows from the Mayer-Vitorisexact sequence. Thus, the property (2) is quite delicate anddoes depend on how subspacesA andB are located in the ambient spaceA ∪ B. With this definition there could be asmany Euler–Poincare characteristicsχH∗ as different ordinary cohomology theoriesχH∗

exist.

In most cases objects treated in geometry and topology are finite CW -complexes orCW -complexes. For these objects, the ordinary cohomology theory turns out to beunique,that is to say,if it is restricted to the category of pairs of finiteCW complexesany otherordinary (co)homology theory is in fact isomorphic to the singular cohomology theory,namely we have Eilenberg–Steenrod Theorem [ES1, ES2]:


Theorem 2.3. On the category of pairs of finiteCW complexes there exists a unique (upto isomorphism) ordinary (co)homology theory.

The uniqueness in fact follows from the Dimension Axiom. Hence, if the DimensionAxiom is dropped, then there are many cohomology theories. Acohomology theory with-out Dimension Axiom being required is calleda generalized cohomology theoryor anextraordinary cohomology theory. Typical ones areK-theory and the cobordism theoryand the latter is the main topic of this note.

Therefore, the Euler–Poincare characteristic of a finiteCW -complex is uniquely de-fined by any ordinary cohomology theory, e.g., the singular cohomology theory:

χ(X) :=∑

i≥0

(−1)i dimR Hi(X; Z) ⊗ R.

However, for non-finiteCW -complexes this Euler–Poincare characteristic is not nec-essarily uniquely defined and furthermore for any ordinary cohomology theory theH∗-Euler–Poincare characteristicχH∗(X) of then-dimensional Euclidean spaceRn is always

χH∗(Rn) = 1 for anyn

sinceRn is homotopic to one point space. Namely, this characteristic does not give riseto the above looked-for cardinalityχtop for open cells. To remedy this, we use thecoho-mology with compact supportor the so-calledBorel–Moore homology groupfor locallycompact spaces, which can be made into compact spaces by adding just one point, i.e.,one-point compactification. The Borel-Moore homology group HBM

∗ (X; R) of a locallycompact Hasudorff spaceX is the relative singular homology of the one-point compactifi-cationX+ with ∗ being the one point attached:

HBM∗ (X; Z) := H∗(X

+, ∗; Z).

With this we have thatHBMn (Rn; Z) = Z andHBM

k (Rn; Z) = 0 for k 6= n. Thus theBorel–Moore homological Euler–Poincare characteristicχBM (X) of a locally compactHasudorff spaceX can be defined by

χBM (X) :=∑

n

(−1)n dimR HBMn (X) ⊗ R.

Then for a finiteCW -complexX we have

χtop(X) = χ(X) = χBM (X) and χtop(Rn) = χBM (Rn) = (−1)n.

It is known (e.g., see [Ha, Corollary 2.24]) that in the category of finiteCW -complexes,for any two subcomplexesA andB of aCW -complexA ∪ B, the triple(A ∪ B; A, B) isproper, we get the following:

Theorem 2.4. On the category of finiteCW -complexesχtop = χBM satisfies the follow-ing properties:

(1) A ∼= B (homeomorphism) impliesχtop(A) = χtop(B),(2) χtop(A ⊔ B) = χtop(A) + χtop(B),(3) χtop(A × B) = χtop(A) · χtop(B),(4) χtop(pt) = 1.

Remark 2.5. SinceA andB are subcomplexes of aCW -complexA ⊔ B, it is automati-cally thatA andB are two connected components.

Summing up roughly:

6 SHOJI YOKURA(∗)

(1) If we stick to the fundamental/basic/essential propertieswhich the way we countfinite sets satisfies even when we “count” or “measure” topological spaces “topo-logically”, we naturally ends up with the so-called Euler number for finiteCW -spaces.

(2) Such a cardinalityχtop, which becomes the Euler–Poincare characteristic on thecategory of finite CW-complexes, is just one aspect of the unique ordinary coho-mology theory.

The Euler–Poincare characteristic is the simplest but most important invariant in mod-ern geometry and topology. There are many important resultsconcerning it, such as theChern’s Gauss–Bonnet Theorem and the Poincare–Hopf Theorem relating vector fields ona manifold and The Euler–Poincare characteristic of the manifold. Before going further on,we recall MacPherson’s Theorem [Mac], which is a theorem of Grothendieck–Riemann–Roch type:

Theorem 2.6. LetF be the covariant functor of constructible functions on the category ofcomplex algebraic varieties with proper morphisms. There exists a unique natural trans-formation from the covariant functorF to the Borel–Moore homology theory

c∗ : F → HBM∗

such that ifX is a smooth variety then the valuec∗(1X) of the characteristic function1X on X is the Poincare dual of the total Chern cohomology classc(TX) of the tangentbundle:

c∗(1X) = c(TX) ∩ [X].

A bivariant-theoretic analogue of the above transformation was conjectured in [FM]and it was solved affirmatively by J.-P. Brasselet under a certain condition [Br], and it wasinvestigated furthermore in [BSY3, BSY4].

In the above we saw that one automatically obtains the notionof Euler–Poincare char-acteristic just by changing the requirement of set isomorphism to that of the topologicalisomorphism in the property (1). Now the notion ofgenusis a “drastic” one along thesame line of thinking. In the following part of this section,all manifolds are assumed to besmooth, compact and oriented.

Rene Thom [Th] made an epoc: he introduced the notion of cobordism or cobordant,i.e., he suggested twon-dimensional manifoldsA, B to be “identified” if there exists an(n + 1)-dimensional manifoldW whose boundary∂W is isomorphic to the disjoint unionof A and−B, i.e.,B with its orientation reversed:

∂W = A ∪−B.

ThenA andB are called “bordant” and denoted by

A ∼= B.

And we consider the following problem of “cardinality” based on the bordism “isomor-phism”:

Problem 2.7. On closed manifolds find a “cardinality”γ satsifying the following condi-tions:

(1) A ∼= B (bordant) impliesγ(A) = γ(B),(2) γ(A ⊔ B) = γ(A) + γ(B),(3) γ(A × B) = γ(A) · γ(B),(4) γ(pt) = 1.


In other words, if we letR be a commutative ring with unit, thenγ is simply a ring homo-morphism from the bordism ringΩ to R:

γ : Ω → R.

This “cardinality” is called a genus. Here we recall thatΩ = ΩSO is the set of (co)bordismclasses of smooth, compact and oriented manifolds, which becomes a ring with additionby the disjoint union of manifolds and multiplication by Cartesian product of manifolds.

Remark 2.8. (1) Any closed manifold cannot be decomposed into closed submanifolds,unless it is already the disjoint union of closed manifolds.Hence in the property (2)A ⊔ B is automaticallythe disjoint union of closed manifolds. (2) In the case of thetopological isomorphism, by decomposing the 1-dimensional Euclidean spaceR1 into(−∞, 0)⊔ 0 ⊔ (0,∞), we get the Euler–Poincare characteristic. However, in the aboveproblem we cannot do such a magic or trick.

In contrast to the Euler–Poincare characteristicχtop (which is certainly not a genus fororiented manifolds, but a genus only for stable complex manifolds), it is not uniquely de-termined. Indeed, the Hirzebruch’s famous signatureσ and A are the most typical andwell-studied genera.

Here is a very simple problem on genera:

Problem 2.9. Determine all genera.

This problem is in a sense solved by the following fundamental theorem due to R.Thom:

Theorem 2.10. On the category of closed oriented manifolds we have

Ω ⊗ Q ∼= Q[P2, P4, P6, · · · , P2n, . . . ].

So, if we consider a commutative ringR without torsion for a genusγ : Ω → R, thenthe genusγ is completely determined by the valueγ(P2n) of the cobordism class of eacheven dimensional complex projective sapceP2n. Then using this value one could considerits generating “function” or formal power series such as

∑

n γ(P2n)xn, or∑

n γ(P2n)x2n,and etc. In fact, a more interesting problem than determining “all genera” is is to determinall rigid genera such as the above mentioned signatureσ andA; namely a genera satisfyingthe following multiplicativity stronger than the product property (2):

(2)rigid : γ(M) = γ(F )γ(B) for a fiber bundleM → B with its fiberF being a spin-manifold and compact connected structural group.

For this rigidity problem on genera, one needs to consider its so-called “logarithmic”formal power series inR[[x]]:

logγ(x) :=∑

n

1

2n + 1γ(P2n)x2n+1.

Now here is Ochanine–Bott–Taubes’ epoc-making theorem:

Theorem 2.11. The genusγ is rigid if and only if it is an elliptic genus, i.e., its logarithmlogγ is an elliptic integral, i.e.,

logγ(x) =

∫ x

0

1√1 − 2δt2 + ǫt4

dt

for someδ, ǫ ∈ R.

Remark 2.12. We note that if one allows its fiberF to be any manifold instead of a spin-manifold, then only the signature is rigid.

8 SHOJI YOKURA(∗)

S. Ochanine [Oc] proved the “only if” part and later the “if part” was first “physically”proved by E. Witten [Wi] using the Dirac operator on the loop spaces and rigorously provedby C. Taubes [Ta] and also by R. Bott and C. Taubes [BT]. Also see B. Totaro’s papers[To-1, To-2].

Given a ring homomorphismϕ : MSO∗(pt) → R, R is aMSO∗(pt)-module and

MSO∗(X) ⊗MSO∗(pt) R

becomes “almost” a generalized cohomology theory in the sense that it does not necessarilysatisfy the Exactness Axiom. P. S. Landweber [La1] gave an algebraic criterion (called theExact Functor Theorem) for it to become a generalized cohomology theory, i.e., satisfy theExactness Axiom too. Applying Landweber’s Exact Functor Theorem, P. E. Landweber,D. C. Ravenel and R. E. Stong [LRS] showed the following theorem:

Theorem 2.13. For the elliptic genus

γ : MSO∗(pt) = MSO∗(pt) = Ω → Z[1

2][δ, ǫ],

the following functors are generalized cohomology theories:

MSO∗(X) ⊗MSO∗(pt) Z[1

2][δ, ǫ][ǫ−1],


2][δ, ǫ][(δ2 − ǫ)−1],


2][δ, ǫ][∆−1],

where∆ = ǫ(δ2 − ǫ)2.

More generally J. Franke [Fr] showed the following theorem:

Theorem 2.14. For the elliptic genus

γ : MSO∗(pt) = MSO∗(pt) = Ω → Z[1

2][δ, ǫ],

the following functor is a generalized cohomology theory:


2][δ, ǫ][P (δ, ǫ)−1],

whereP (δ, ǫ) is a homogeneous polynomial of positive degree withdeg δ = 4, deg ǫ = 8.

The generalized cohomology theory


2][δ, ǫ][P (δ, ǫ)−1]

is called anelliptic cohomology theory. It is defined in an algebraic manner, but not in amore topological or geometric manner as in K-theory, the bordism theoryMSO∗(X). So,people have been searching for a reasonable geometric or topological construction of theelliptic cohomology (e.g., see [KS] by M. Kreck and S. Stolz). In fact, the present workis motivated partly by this interesting geometric problem of the elliptic cohomology theory.

Remark 2.15. Before finishing this section, as to the notion of “cardinality” we would liketo remark about a recent work of J. Baez and W. Dolan [BD] on what they call “homotopycardinality” from the aspect of “categorification”. The above Euler–Poincare characteristicof a topological spaceX is the alternating sum of the dimension of (co)homology groupsH∗(X; R) or H∗(X; R). Baez–Dolan’s homotopy cardinality of a topological spaceX,


denoted byχπtop(X) provisionally here, is a sort of “alternating product” of homotopy

groups ofX; to be more precise,

χπtop(X) :=

∏

k≥0 |π2k(X)|∏

k≥0 |π2k+1(X)| .

Where|πi(X)| denotes the cardinality of the groupπi(X) considered as a category, i.e., agroupoid. This homotopy cardinality makes sense of course when it is well-defined, justas it is the case for the Euler–Poincare characteristic. Baez and Dolan showed that thishomotopy cardinality satisfies the same property as the Euler–Poincare characteristic. Sofar it is a reasonable cardinality only when all the homotopygroups are finite groups andno other case has not been computed and their conjecture is that there would be no othercase than finite homotopy groups. It would be an interesting problem whether there is areasonable relation between the Euler–Poincare characteristic and the Baez–Dolan homo-topy cardinality. We hope to come back to this problem. For more details see their paper[BD] and also Baez’s homepage (http://math.ucr.edu/home/baez/). In particular, one is rec-ommended to read Baez’s articles [Ba1, Ba2] and also T. Leinster’s paper [Lein].

3. FULTON–MACPHERSON’ S BIVARIANT THEORY

We make a quick review of Fulton-MacPherson’s bivariant theory [FM], where thereader also will find a lot of examples like algebraic K-theory and operational Chow groupsin algebraic geometry, or bivariant versions of generalized cohomology theories in alge-braic topology.Let V be a category with a final objectpt, a class of “indepedent squares” and a classCof “confined maps”, which is closed under composition and base change in indepedentsquares and contains all identity maps. Here we assume that all indepedent squares arefiber products (i.e. the corresponding fiber products exist in V), which satisfy the follow-ing properties:

(i) if the two inside squares in

X′′ h′

−−−−→ X′ g′

−−−−→ X

yf ′′

yf ′

y

f

Y ′′ −−−−→h

Y ′ −−−−→g

Y

or

X′ −−−−→h′′

X

f ′

y

y

f

Y ′ −−−−→h′

Y

g′

y

y

g

Z′ −−−−→h

Z

are independent, then the outside square is also independent,

10 SHOJI YOKURA(∗)

(ii) any square of the following forms are independent:

X

f

idX// X

f

X

idX

f// Y

idY

YidX

// Y Xf

// Y

wheref : X → Y is anymorphism.

Example 3.1. Examples for “confined maps” are proper or projective maps inalgebraictopology or geometry. In the category of smooth manifolds and maps, independent squaresare by definition the squares withf andg transversalin the usual sense, i.e.f × g istransversal to the diagonal submanifold ofY ×Y so that the fiber productX′ = Y ′×Y X

exists as a smooth manifold. Note that in this category not all fiber products exist.

A bivariant theoryB on a categoryV with values in the category of graded abeliangroups is an assignment to each morphism

Xf−→ Y

in the categoryV a graded abelian group (in most cases we ignore the grading )

B(Xf−→ Y )

which is equipped with the following three basic operations. The i-th component of

B(Xf−→ Y ), i ∈ Z, is denoted byBi(X

f−→ Y ).

Remark 3.2. One can also considerZ2-grading and0-grading, i.e., no grading.

Product operations: For morphismsf : X → Y andg : Y → Z, the product operation

• : Bi(Xf−→ Y ) ⊗ Bj(Y

g−→ Z) → Bi+j(Xgf−→ Z)

is defined.Pushforward operations: For morphismsf : X → Y andg : Y → Z with f confined,

the pushforward operation

f∗ : Bi(Xgf−→ Z) → Bi(Y

g−→ Z)

is defined.Pullback operations: For anindependentsquare

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y,

the pullback operation

g∗ : Bi(Xf−→ Y ) → Bi(X′ f ′

−→ Y ′)

is defined.And these three operations are required to satisfy the sevencompatibility axioms (see

[FM, Part I,§2.2] for details):(B-1) product is associative,(B-2) pushforward is functorial,(B-3) pullback is functorial,(B-4) product and pushforward commute,


(B-5) product and pullback commute,(B-6) pushforward and pullback commute, and(B-7) projection formula.

We also assume thatB has units:Units: B has units, i.e., there is an element1X ∈ B0(X

idX−−→ X) such thatα • 1X = α

for all morphismsW → X, all α ∈ B(W → X); such that1X • β = β for all morphismsX → Y , all β ∈ B(X → Y ); and such thatg∗1X = 1X′ for all g : X′ → X.

Let B, B′ be two bivariant theories on a categoryV. Then aGrothendieck transformationfrom B to B′

γ : B → B′

is a collection of homomorphisms

B(X → Y ) → B′(X → Y )

for a morphismX → Y in the categoryV, which preserves the above three basic opera-tions:

(i) γ(α •B β) = γ(α) •B′ γ(β),(ii) γ(f∗α) = f∗γ(α), and(iii) γ(g∗α) = g∗γ(α).

A bivariant theory unifies both a covariant theory and a contravariant theory in the fol-lowing sense:

B∗(X) := B(X → pt) becomes a covariant functor forconfinedmorphisms and

B∗(X) := B(Xid−→ X) becomes a contravariant functor foranymorphisms.

And a Grothendieck transformationγ : B → B′ induces natural transformationsγ∗ :B∗ → B′

∗ andγ∗ : B∗ → B′∗.

As to the grading,Bi(X) := B−i(Xid−→ X) andBj(X) := Bj(X

id−→ X).

A bivariant theory is calledcommutative(see [FM,§2.2]) if whenever both

W

f ′

g′

// X

f

W

g′

f ′

// Y

g

Y g// Z X

f// Z

are independent squares, then forα ∈ B(Xf−→ Z) andβ ∈ B(Y

g−→ Z)

g∗(α) • β = f∗(β) • α.

If g∗(α) • β = (−1)deg(α) deg(β)f∗(β) • α, then it is calledskew-commutative.

4. BOREL–MOORE FUNCTOR IN THE SENSE OFLEVINE–MOREL

For some special classes of morphisms an additive bivarianttheory (see below) carriesthe so-called Borel–Moore functor with products, which is the basic ingredient for Levine–Morel’s construction of algebraic cobordism [LM3].

Definition 4.1. ([FM, 2.6.2 Definition, Part I]) LetS be a class of maps inV, which isclosed under composition and containing all identity maps.Suppose that to eachf : X →Y in S there is assigned an elementθ(f) ∈ B(X

f−→ Y ) satisfying that

(i) θ(g f) = θ(f) • θ(g) for all f : X → Y , g : Y → Z ∈ S and


(ii) θ(idX) = 1X for all X with 1X ∈ B∗(X) := B(XidX−−→ X) the unit element.

Thenθ(f) is called acanonical orientationof f .

For a morphismf : X → Y ∈ S theGysin homomorphism

f ! : B∗(Y ) → B∗(X) defined by f !(α) := θ(f) • α

is contravariantly functorial. And for the following independent square

Xf−−−−→ Y

idX

y

yidY

X −−−−→f

Y,

wheref ∈ C ∩ S, theGysin homomorphism

f! : B∗(X) → B∗(Y ) defined by f!(α) := f∗(α • θ(f))

is covariantly functorial. The notation should carry the information ofS and the canonicalorientationθ, but it will be usually omitted if it is not necessary to be mentioned.

Definition 4.2. (i) Let S be another class of maps inV , called “specialized maps” (e.g.,smooth maps in algebraic geometry), which is closed under composition and under basechange in independent squares and containing all identity maps. LetB be a bivariant theory.If S has canonical orientations inB, then we say thatS is canonicalB-orientable and anelement ofS is called a canonicalB-orientable morphism. If we fix the orientation, thenwe say thatS is canonicalB-oriented. (Of courseS is also a class of confined maps, butsince we consider the above extra condition ofB-orientability onS, we give a differentname toS.)

(ii) Assume thatS is canonicalB-oriented. Furthermore, if the orientationθ on Ssatisfies that for an independent square withf ∈ S

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y

the following condition holds

θ(f ′) = g∗θ(f),

(which means that the orientationθ preserves the pullback operation), then we callθ anicecanonical orientationand say thatS is nicely canonicalB-orientedand an element ofS iscalleda nicely canonicalB-oriented morphism.

Example 4.3. The following are typical examples of nice canonical orientations:

(1) Smooth morphisms in algebraic geometry for bivariant algebraic K-theory andoperational Chow groups [FM, p.28–29].

(2) Smooth oriented submersions in the category of smooth manifolds for the bivari-ant versions of generalized cohomology theories, which have “canonical Thom-classes” for oriented vector bundles [FM, p.46, p.49]. Examples of the generalizedcohomology theories are usual cohomologyH∗, real K-theoryKO[ 12 ] with 2 in-verted, or oriented cobordismMSO∗.

In the following proposition we assume that for all morphismf : X → X′ andg : Y →Y ′ in V any of the four small squares in the big diagrams below are independent (hence


any square is independent):

X × Yf×IdY−−−−→ X′ × Y

p×IdY−−−−→ Y

IdX×g

y

y

IdX′×g

y

g

X × Y ′ f×IdY ′−−−−−→ X′ × Y ′ p×IdY ′−−−−−→ Y ′

IdX×q

y

y

IdX′×q

y

q

X −−−−→f

X′ −−−−→p

pt.

Proposition 4.4. Let B be a bivariant theory and letS be a class of specialized mapswhich isnicelycanonicalB-oriented.

(1-i) For an independent square

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y

with g ∈ C andf ∈ S, the following diagram commutes:

B∗(Y′)

f ′ !

−−−−→ B∗(X′)

g∗

y

yg′

∗

B∗(Y ) −−−−→f !

B∗(X),

(1-ii) The covariant functorB∗ for confined morphisms and the contravariant functorB∗ for morphisms inS are both compatible with the exterior product

× : B(X −→ pt) ⊗ B(YπY−−→ pt) → B(X × Y → pt)

defined byα × β := π∗

Y α • β.

(2-i) For an independent square

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y

with g ∈ C ∩ S, the following diagram commutes:

B∗(Y ′)f ′∗

−−−−→ B∗(X′)

g!

y

yg′

!

B∗(Y ) −−−−→f !

B∗(X),

(2-ii) The contravariant functorB∗ for anymorphisms and the covariant functorB∗ formorphisms inC ∩ S are both compatible with the exterior product

× : B(XidX−−→ X) ⊗ B(Y

idY−−→ Y ) → B(X × YidX×X−−−−→ X × X)

defined byα × β := p1

∗α • p2∗β


wherep1 : X × Y → X andp2 : X × Y → Y be the projections.

Remark 4.5. The covariant and contravariant functorsB∗ andB∗ arealmostwhat Levineand Morel callBorel–Moore functor with productsin [LM3, Mer]; namely they do notnecessarily have theadditivity property, which is, for example, the following one in thecase of homology theory of topological spaces: for the disjoint unionX ⊔ Y of spaces

H∗(X ⊔ Y ) = H∗(X) ⊕ H∗(Y ).

If we want a bivariant theory to have such an additivity property, we need a bit more re-quirements on the category, but here we do not go into details(see [Yo1] for more details),since the additivity is not an essential ingredient.

5. A UNIVERSAL BIVARIANT THEORY

We have constructed a universal or “motivic” bivariant one among certain bivarianttheories [Yo1].

Theorem 5.1. Let V be a category with independent squares, a classC of confined mapsand a classS of specialized maps as before. Assume that any Cartesian diagram withf

specialized exists inV and is independent withf ′ also specialized. We define

MCS(X

f−→ Y )

to be the free abelian group generated by the set of isomorphism classes of confined mor-phismsh : W → X such that the composite ofh andf is a specialized map:

h ∈ C and f h : W → Y ∈ S.

(1) The associationMCS is a bivariant theory with the nice canonical orientationθ(f) :=

[idX ] for f ∈ S, if the three operations are defined as follows:Product operations: For morphismsf : X → Y andg : Y → Z, the product operation

• : MCS(X

f−→ Y ) ⊗ MCS(Y

g−→ Z) → MCS(X

gf−→ Z)

is defined by(

∑

V

mV [VhV−−→ X]

)

•(

∑

W

nW [WkW−−→ Y ]

)

:=∑

V,W

mV nW [V ′ hV kW′′

−−−−−−→ X],

where we consider the following fiber squares

V ′h′

V−−−−→ X′ f ′

−−−−→ W

kW′′

ykW

′

ykW

y

V −−−−→hV

X −−−−→f

Y −−−−→g

Z.

Pushforward operations: For morphismsf : X → Y andg : Y → Z with f confined,the pushforward operation

f∗ : MCS(X

gf−→ Z) → MCS(Y

g−→ Z)

is defined by

f∗

(

∑

V

nV [VhV−−→ X]

)

:=∑

V

nV [VfhV−−−→ Y ].


Pullback operations: For an independent square

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y,


g∗ : MCS(X

f−→ Y ) → MCS(X′ f ′

−→ Y ′)

is defined by

g∗

(

∑

V

nV [VhV−−→ X]

)

:=∑

V

nV [V ′ hV′

−−→ X′],

where we consider the following fiber squares:

V ′ g′′

−−−−→ V

hV′

y

yhV

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y.

(2) LetB be a bivariant theory onV such thatS is nicely canonicalB-oriented. Thenthere exists a unique Grothendieck transformation

γB : MCS → B

such that for a specialized morphismf : X → Y ∈ S the homomorphismγB : MCS(X

f−→Y ) → B(X

f−→ Y ) satisfies the normalization condition that

γB([XidX−−→ X]) = θB(f).

Corollary 5.2. (1) The abelian groupMCS∗

(X) := MCS(X → pt) is the free abelian group

generated by the set of isomorphism classes

[VhV−−→ X]

such thathV : V → X ∈ C andV → pt is a specialized map inS.

(2) The abelian groupMCS

∗(X) := MC

S(XidX−−→ X) is the free abelian group generated

by the set of isomorphism classes

[VhV−−→ X]

such thathV : V → X ∈ C ∩ S.(3) Both functorMC

S∗andMC

S

∗are Borel–Moore functors with products, except for the

additivity property.

Remark 5.3. The assumptions of Theorem 5.1 are satisfied in the context ofExample4.3 withS the class of smooth morphisms or oriented submersions. So inExample 4.3MC

S∗(X) (resp. MC

S

∗(X)) is the free abelian group generated by isomorphism classes

of proper mapsV → X with V smooth or an oriented manifold (resp. of proper mapsh : V → X with h smooth or an oriented submersion).


Remark 5.4. A more subtle ring similar to the abelian groupMCS(X

f−→ Y ) in algebraicgeometry is the so-called relative Grothendieck ring or “motivic ring”, which was intro-duced by E. Looijenga [Lo] and furthermore studied by F. Bittner [Bit]. This plays animportant role in the motivic measure and integration (e.g.see [DL1, DL2, DL3, Ve]) andalso for the motivic characteristic classes for singular varieties [BSY1, BSY2, SY].

For more details and an abstract oriented bivariant theory see [Yo1] and in [Yo2] we willdeal with a moregeometricaloriented bivariant theory, i.e., a bivariant-theoretic version ofLevine–Morel’s algebraic cobordism.

6. BIVARIANT BORDISM THEORY FOR SMOOTH MANIFOLDS

Now in this section we want to apply our universal bivariant theory to cobordism theory.More precisely we want to get a bivariant bordism group in such a way that its associatedcovariant functor is supposed to be the bordism groupMSO∗(X). For that purpose we firstintroduce or recall the following definition of a parameterized family of smooth manifolds:

Definition 6.1. (i) In the topological contextV is the category of locally compactHausdorff spaces, which can be embedded as a closed subset into someRn [FM,p.32]. Confined maps and independent squares are defined to beproper maps andfiber products, respectively. Afamily h : M → X of compact manifolds (withboundary)is a proper continous maph, which locally onM is isomorphic to theprojection onto the first factor

U ≃ h(U) × V 7→ h(U) ,

with V an open subset of(x1, . . . , xn) ∈ Rn| x1 ≥ 0 (see [FM, p.65]).(ii) In the smooth contextV is the category of finite dimensional smooth manifolds

and smooth maps between them. Confined maps and independent squares aredefined to be proper maps andtransversalfiber products (as before). Afamilyh : M → X of smooth compact manifolds (with boundary)is a proper smoothmaph, which locally onM is isomorphic to the projection onto the first factor

U ≃ h(U) × V 7→ h(U) ,

with V an open subset of(x1, . . . , xn) ∈ Rn| x1 ≥ 0. In other wordsh is asubmersion.

(iii) Moreover we assume that a consistentorientationof the fiber manifolds has beenchosen, e.g. one has a coordinate coveringU ≃ h(U) × V 7→ h(U) of M suchthat the corresponding change of coordinates is given by orientation preservinghomeomorphisms (or diffeomorphisms) between open subsetsof (x1, . . . , xn) ∈Rn| x1 ≥ 0.

(iv) Note that thefiberwise boundary ofM , denoted by∂h(M) (corresponding to thepoints withx1 = 0 in the above coordinates), with the induced map∂h(M) → X

is again a “family of compact (smooth) oriented manifolds (without boundary)”.

For a “family h : M → X of compact (smooth) oriented manifolds (with boundary)”its reverse oriented family is denoted by−h : −M → X.

Remark 6.2. The above “familyh : M → X of compact (smooth) oriented manifolds(with boundary)” is stable under base change (in independent squares) and composition (ifat most one fiber of the two families may have a boundary). The usual notion of a fiberbundle overX, with fiber a compact (smooth) oriented manifoldF (with boundary) andstructure group the homeomorphisms (or diffeomorphisms) of the fiberF , is certainly sta-ble under base change, but it is not clear whether or not it is also stable under composition.


Originally we tried to formulate our bivariant bordism theory for smooth manifolds usingfiber bundles and we came across to this stability problem. Soto overcome or to avoid thisunstability problem of fiber bundles we use such families above.

Definition 6.3. Let h : M → X, h′ : M ′ → X′ be two families of compact (smooth)oriented manifolds without boundary. If there exists a family H : W → X of com-pact (smooth) oriented manifolds with boundary such that the induced familyH∂H(W) :∂H(W ) → X is isomorphic to

h + (−h) : M ⊔ (−M ′) → X ,

then the two familiesh : M → X, h′ : M ′ → X are calledfiberwise bordant. The

fiberwise bordism class of such a familyh : M → X is denoted by[M h−→ X]fib.

Lemma 6.4. The fiberwise bordism is an equivalence relation for families over the samebase space.

Definition 6.5. Let a morphismf : X → Y and two morphismsh : M → X, h′ : M ′ →X′ in the categoryV as above (i.e. either in the topological or smooth context) be givensuch that the compositef h andf h′ are both a family of compact (smooth) orientedmanifolds without boundary. Then these are calledfiberwise bordant with respect tof , orsimply f-fiberwise bordant, if there exists a morphismH : W → X such thatf h isa family of compact (smooth) oriented manifolds with boundary with the induced familyf H∂fH (W) : ∂fH(W ) → X isomorphic to

f h + (−(f h′)) : M ⊔ (−M ′) → X .

Lemma 6.6. The fiberwise bordism with respect to a morphismf is an equivalence rela-tion.

This equivalence relation is calledf-fiberwise bordismand thef-fiberwise bordism

class ofh : M → X with respect to a morphismf is denoted by[Mh−→ X]fib

f . For

the identity mapidX : X → X, the idX-fiberwise bordism class[M h−→ X]fibidX

is the

fiberwise bordism class[Mh−→ X]fib. For a constant mapπX : X → pt theπX-fiberwise

bordism class[M h−→ X]fibπX

is simply denoted by[M h−→ X].

Definition 6.7. (Relative Fiberwise Bordism Group) For a morphismf : X → Y in V,

FΩ−n(Xf−→ Y )

denotes the set off-fiberwise bordism classes[Mh−→ X]fib

f such thatf h : M → Y isa family of compact oriented (smooth) manifolds without boundary, whose fibers are puren-dimensional. Of course we assume that the fibers of a corresponding cobordism are pure

n +1-dimensional compact oriented (smooth) manifolds with boundary.FΩ−n(Xf−→ Y )

is an abelian group with the addition coming from disjoint union:

[M1h1−→ X]fib

f + [M2h2−→ X]fib

f := [M1 ⊔ M2h1+h2−−−−→ X]fib

f .

The unit is[∅ → X]fibf and[−M

−h−−→ X]fibf = −[M

h−→ X]fibf .

For a mapπX : X → pt to a point, the relative fiberwise bordism group is the classicalbordism group oriented topological (or smooth)n-dimensional manifolds proper overX:

FΩ−n(XπX−−→ pt) = MSO(top)

n (X).

For the identity mapidX : X → X, we set

FMSO−n(X) := FΩ−n(XidX−−→ X).


It is obvious by definition thatMSO∗(X) is a covariant functor andFMSO∗(X) is acontravariant functor.FMSO∗(X) shall be called thefiberwise cobordism group ofX.The notationFMSO∗ is adopted to avoid some possible confusion with the usual cobor-dism groupMSO∗(X).

Theorem 6.8. We consider either the topological or smooth context with the underlyingcategoryV as before. Then therelative fiberwise bordism groups

FΩ−n(Xf−→ Y )

define a (graded) bivariant theory if the three operations are defined as follows:Product operations: For morphismsf : M → N andg : N → Z, the product opera-

tion

• : FΩ−n(Mf−→ N) ⊗ FΩ−m(N

g−→ Z) → FΩ−n−m(Mgf−→ Z)

is defined by(

∑

V

mV [VhV−−→ M ]fib

f

)

•(

∑

W

nW [WkW−−→ N ]fib

g

)

:=∑

V,W

mV nW [V ′ hV kW′′

−−−−−−→ M ]fibgf ,

where we consider the following fiber squares

V ′h′

V−−−−→ M ′ f ′

−−−−→ W

k′′

W

yk′

W

ykW

y

V −−−−→hV

M −−−−→f

N −−−−→g

Z.

Pushforward operations: For morphismsf : M → N andg : N → Z with f beingproper, the pushforward operation

f∗ : FΩ−n(Mgf−→ Z) → FΩ−n(N

g−→ Z)

is defined by

f∗

(

∑

V

nV [VhV−−→ M ]fib

gf

)

:=∑

V

nV [VfhV−−−→ N ]fib

g .

Pullback operations: For an independent square

M ′ g′

−−−−→ M

f ′

y

y

f

N ′ −−−−→g

N,


g∗ : FΩ−n(Mf−→ N) → FΩ−n(M ′ f ′

−→ N ′)

is defined by

g∗

(

∑

V

nV [VhV−−→ M ]fib

f

)

:=∑

V

nV [V ′ h′

V−−→ N ′]fibf ′ ,


where we consider the following fiber squares:

V ′ g′′

−−−−→ V

h′

V

y

yhV

M ′ g′

−−−−→ M

f ′

y

y

f

N ′ −−−−→g

N.

Remark 6.9. In other words the operations above are induced from the correspondingoperations in Theorem 5.1 by dividing out the fiberwise bordism relation. The main pointis then to show that these operations are welldefined, i.e. are independent of the choicesfor the representing morphisms. For example for the bivariant product

[VhV−−→ M ]fib

f • [WkW−−→ N ]fib

g = [V ′ hV k′′

W−−−−−→ M ]fibf

one shows first that for fixedkW : W → N the righthand side only depends on thef-fiberwise bordism classof hV : V → M . For this one looks at the correponding fibersquares with a corresponding bordism substituted forhV . Then one shows in the same waythat the righthand side only depends on theg-fiberwise bordism classof kW : W → N .Of course here we need the stability properties mentioned atthe beginning of this sectionfor our notion of a family of compact oriented (smooth) manifolds (with boundary).

Proposition 6.10. Let FM be the class of families of compact oriented (smooth) man-ifolds (without boundary) in the underlying categoryV. ThenFM is nicely canonicalFΩ-oriented by

θ(f) := [MidM−−→ M ]fib

f ∈ FΩ(Mf−→ X)

for f : M → X a morphism inFM.

Let B be a bivariant theory onV (i.e. in the topological or smooth context) such thatFM is canonicalB-oriented. ThenB together with this orientationθ is called afiberwisebordism invariant oriented bivariant theory, if

h∗(θ(f h)) = h′∗(θ(f h′))

for any morphismf : X → Y and any two proper morphismsh : M → X, h′ : M ′ → X′

in the categoryV such that the compositef h andf h′ in FM aref-fiberwise bordant.In other words, thef-fiberwise bordism class

[Mh−→ X]fib

f ∈ FΩ(Xf−→ Y )

uniquely determines the element

h∗(θ(f h)) ∈ B(Xf−→ Y ) .

Now we can state a universality theorem for the bivariant relative fiberwise bordismtheoryFΩ.

Theorem 6.11. Let B be a bivariant theory onV such thatFM is nicely canonicalB-oriented. Assume thatB together with this orientationθ is fiberwise bordism invariantoriented. Then there is a unique Grothendieck transformation

γB : FΩ → B


such that for anyf : M → N in FM the homomorphismγB satisfies the normalization

γB([MidM−−→ M ]fib

f ) = θ(f) .

Remark 6.12. Examples of fiberwise bordism invariant oriented bivarianttheories andmore work on the fiberwise cobordism groupFMSO∗(X), in particular related to Kreck–Stolz’s work [KS], will be worked out somewhere else. On the correspondingcovarianttheories one has some natural transformations commuting with exterior products, comingfrom the fundamental class of an oriented (smooth) manifold:

(i) FΩ(X → pt) = MSOtop(X) → H∗(X; Z) in the topological context.(ii) FΩ(X → pt) = MSO(X) → MSOtop(X) → H∗(X; Z) in the smooth context.(iii) FΩ(X → pt) = MSO(X) → KO(X)[ 1

2] in the smooth context.

And these can at least be extend to a suitable Grothendieck transformation on a bivariantsubtheory ofFΩ by the results of [BSY4].

Acknowledgements.The present work was initiated during my stay in August 2007 atthe Erwin Schrodinger International Institute of Mathematical Physics (ESI), Vienna, andthe International Center of Theoretical Physics (ICTP), Trieste. I would like to thank thestaff of both institutes, in particular, Professor Le D. T.(ICTP) and Professor Peter Michor(Universitat Wien and ESI), for their hospitality and for anice atmosphere in which towork. I would also like to thank the organizers of the Fourth Franco-Japanese Symposiumon Singularities (Toyama, Japan, August 27 - 31, 2007) for iniviting me to contribute apaper to the proceedings of the symposium. Finally I would like to thank Jorg Schurmannand the referee for their valuable comments and suggestions.

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A Universal Bivariant Theory and Cobordism Groups · A UNIVERSAL BIVARIANT THEORY AND COBORDISM GROUPS SHOJI YOKURA(∗) Dedicated to Professor Mutsuo Oka on the occasion of his 60th