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APPEARED IN BULLETIN OF THE

arXiv:math/9204231v1 [math.AT] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 304-309A COMBINATORIAL FORMULAFOR THE PONTRJAGIN CLASSESI. M. Gelfand and R. D. MacPhersonAbstract.

A combinatorial formula for the Pontrjagin classes of a triangulatedmanifold is given. The main ingredients are oriented matroid theory and a modifiedformulation of Chern-Weil theory.1.

IntroductionThe problem of finding a combinatorial formula for the Pontrjagin classes of apolyhedral manifold X is forty-five years old and has stimulated much research (see[M] and [L] for references). Chern-Weil theory provides a formula for the Pontrjaginclasses of a Riemannian manifold: They are represented by differential forms thatmeasure certain types of curvature of the manifold.The problem is to find ananalogous theory for polyhedra, which have “infinite curvature” at the corners.In this note, we announce a formula that holds in all dimensions, is com-pletely explicit, and can be calculated using combinatorial constructions and theoperations of finite-dimensional linear algebra over Q.For each i, the formulagives a rational simplicial cycle ζi in the barycentric subdivision of X, whosePoincar´e dual represents the ith inverse Pontrjagin class ˜pi(X).

(The inversePontrjagin classes of X are defined from the usual Pontrjagin classes pi(X) by(1 + p1(X) + p2(X) + · · · ) ⌣(1 + ˜p1(X) + ˜p2(X) + · · · ) = 1. Like the pi, the˜pi generate the Pontrjagin ring.) The cycle ζi depends on the choice of certainadditional combinatorial data called a fixing cycle.

We think of a fixing cycle as acombinatorial analogue of a smooth structure on X. In fact, a smooth structure onX induces a canonical fixing cycle.Two ideas make this formula possible: the systematic exploitation of orientedmatroids and a reformulation of Chern-Weil curvature theory.The only other general and explicit combinatorial formula for the Pontrjaginclasses is Cheeger’s [C].

It uses the asymptotics of the spectrum of a differentialoperator, so it is difficult to compute and its rationality properties are not clear.However, Cheeger’s formula is clearly the best one for the context of the Hodgeoperator constructed from the metric.1991 Mathematics Subject Classification. Primary 55R40, 57Q50, 57R05.Received by the editors January 30, 1990 and, in revised form, August 20, 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2I. M. GELFAND AND R. D. MACPHERSON2.

Review of oriented matroidsOriented matroid theory is a well-developed branch of combinatorics with manyapplications. One of our goals is to establish a link between this theory and differ-ential topology.

We include here only the definitions from oriented matroid theorythat we need. The best general reference is [BLSWZ].Definition.

An oriented matroid M is a finite set V called the elements of Mtogether with a finite collection of maps ci: V →{−, 0, +} called the covectors ofM subject to the following axioms:1. The constant function with value 0 is a covector.2.

If c is a covector, then −c is a covector where −(−) = +, −(0) = 0, and−(+) = −.3. If c and d are covectors, then c ◦d is a covector where c ◦d is defined byc ◦d(v) = c(v)if c(v) ̸= 0,d(v)otherwise.4.

For all covectors c and d, if v is an element such that c(v) = + and d(v) = −,then there exists a covector e such that• e(v) = 0;• If c(w) = d(w) = 0, then e(w) = 0;• If c(w) ̸= −, d(w) ̸= −but c(w) and d(w) are not both 0, thene(w) = +;• If c(w) ̸= +, d(w) ̸= + but c(w) and d(w) are not both 0, thene(w) = −.The idea behind this definition is that an oriented matroid is a combinatorialabstraction of a finite set V of vectors (which are not assumed to be distinct or tobe nonzero) in a finite-dimensional real vector space W. The covectors c correspondto linear functionals c: W →R, but c(v) remembers only whether c(v) is negative,zero, or positive.An element v of V is nonzero if c(v) ̸= 0 for some covector c.A subset{v1, v2, . .

. , vj} of V is said to be independent if there exists a set of covectors{c1, c2, .

. .

, cj} such that ci(vk) = 0 if and only if i ̸= k. The rank of x is thecardinality of any (and hence every) maximal independent subset of V . The convexhull of a set S of elements is {v ∈V |−∈c(S) if c(v) = −}.

Suppose N and M aretwo matroids with the same set of elements V . We say that N is a strong quotientof M, symbolized M ⇒N, if every covector of N is a covector of M. If M andN have the same rank, we say that the matroid N is a weak specialization of M,symbolized M ⇝N if every covector of N is obtained from some covector of M bysetting nonzero values equal to zero.3.

The formulaLet X be a simplicial manifold of dimension n. For simplicity, we assume thatX is oriented (with orientation class [X]) and that n is odd. The modificationsnecessary for the general case are noted at the end.Definition.

The associated complex Z of X is the simplicial complex constructedas follows: The vertices of Z are quadruples (∆, t, y, z) where• ∆⊂V is a simplex of X, where V is the set of vertices of X.

A COMBINATORIAL FORMULA FOR THE PONTRJAGIN CLASSES3• t, y, and z are oriented matroids of rank n+1, 2, and 1 whose set of elementsis V .• The matroid t has a covector that does not take the value−on any elementof V .• The simplex ∆is related to the matroid t by the following two conditions:1. The nonzero elements of t are exactly the vertices of the star St ∆of ∆.2.

For each simplex ∆′ in St ∆, let V (∆′) be the set of vertices of ∆′. ThenV (∆′) is linearly independent in t and the set of nonzero elements of t inthe convex hull of V (∆′) is just V (∆′) itself.• We have strong quotients t ⇒y ⇒z.The k-simplices of Z are diagrams of weak specializations and inclusionst0⇝t1⇝· · ·⇝tk⇓⇓⇓y0⇝y1⇝· · ·⇝yk⇓⇓⇓z0⇝z1⇝· · ·⇝zk∆0⊆∆1⊆· · ·⊆∆kIf we delete the matroids z resp.

all matroids (t, y, and z) in this definition,we get additional associated simplicial complexes, which we denote by Y resp. eX,equipped with simplicial maps Zρ→Yπ→eX.

Note that eX is just the barycentricsubdivision of X.Remarks. The matroids t are combinatorial abstractions of TpX ⊕R where TpXis the tangent space to X at a point p in the simplex ∆.

The map Yπ→eX is ananalogue of the Grassmannian bundle of two-dimensional quotients of T X ⊕R, andZρ→Y is the circle bundle of one dimensional quotients of the two plane.Proposition 1. The map ρ: Z →Y is topologically a fibration with a circle asfiber.Idea of proof.

That the fibers over the vertices of Y are circles is a special case ofthe Folkman-Lawrence representation theorem for the oriented matroid y.We now give a combinatorial formula for the first Chern class of a triangulatedcircle bundle. Let O be the local system on Y with fiber Q and twisting given by thefiber orientation of Z. Define a 1-cocycle Θ on Z with coefficients in ρ⋆O as follows:For each vertex v of Y , Θ|ρ−1v is the class that integrates to 1 around the circleand has the same value on each 1-simplex.

Having fixed this, for each edge e of Y ,Θ|ρ−1e is the cocycle such that the sum of the squares of the coefficients is minimum.It is rational, since the problem of minimizing a quadratic expression subject tolinear constraints (the cocycle condition) can be solved by linear equations. Now,define the 2-cocycle Ωon Y with coefficients in O by ρ⋆Ω= δΘ.Proposition 2.

The cohomology class {Ω} is the first Chern class of the circlebundle Z.Proof. By construction, {Ω} is the δ2 differential (transgression) of the fiber orien-tation in the spectral sequence for Z.

4I. M. GELFAND AND R. D. MACPHERSONDefinition.

A fixing cycle for X is a (3n −2)-cycle φ ∈Z3n−2(Y, Z) such thatπ⋆(Ωn−1 ⌢φ) = [X].Remark. A fixing cycle is the combinatorial analogue of an orientation class of theGrassmannian bundle.

Unfortunately, it is not unique, as its differential analogueis. The idea evolved from the configuration data of [GGL] and [M].Theorem 1.

Let φ be a fixing cycle for X. Then˜pi(X) ⌢[X] = (−1)iπ⋆(( 12Ω)n+2i−1 ⌢φ).Note that since n is odd, Ωn+2i−1 is a cocycle with coefficients in Q sinceOn+2i−1 = Q.Remark. This is a cycle level formula, since the operations in simplicial (co)ho-mology involved in the right-hand side (π⋆, ⌣, ⌢), and in the definitions of Ω, ρ⋆and δ are all chain level operations.

The complexity of the formula is contained inthe formulas for these operations and the construction of the simplicial complexesZ and Y . Given the fixing cycle φ, it is a purely local formula: the value in anopen set U of eX depends only on the combinatorial structure of X inside U andon φ|π−1U.4.

Construction of the fixing cycleFirst, we study the structure of the auxiliary complex Y . For any simplex ∆in X,let U∆be the set of diagrams of oriented matroids y ⇒t with t satisfying conditions1 and 2 of Definition 1 with respect to ∆.

This is a poset by the specialization (⇝)ordering on the oriented matroids in U∆. We denote the order complex of a posetP by Cx P. The open dual cell of ∆in eX is denoted D∆(so eX is the disjoint unionof the D∆).Proposition 3.

The subcomplex of Y lying over D∆is canonically homeomorphicto Cx U∆× D∆.We denote the homeomorphism of the proposition by c∆: Cx U∆× D∆→Y .Remark. Suppose ∆⊂∆′.

Then the edge of c(C × U∆× D∆) is glued to c(C ×U∆′ ×D∆′) by the map C×U∆→C×U∆′ induced by the map of posets U∆→U∆′defined on the matroid level by setting all elements of V in St ∆but not in St ∆′equal to zero.A smooth structure on X is a homeomorphism σ: X →M to a smooth manifoldthat is differentiable on each closed simplex in X. Define Y to be the Grassmannianbundle whose fiber over x ∈M is the space of n−1-dimensional subspaces F n−1 ⊂TxM ⊕R. The map Yπ→eX is a sort of a “combinatorial model” for the mapY →M.Let Y∆be the part of Y lying over σ(∆) for a simplex ∆⊂X.

Any point yin Y∆determines an element of U∆as follows: Let σ(x) be the image of y in M.There is a unique embedding e: St ∆֒→TxM that is linear on each simplex, takesx to 0, and satisfies d(e|∆′) = d(σ|∆′) for each simplex ∆′ in St ∆. Now map Vinto TxM ⊕R by using the embedding St ∆e→TxM×1→TxM ⊕R for vertices inSt ∆and mapping all other vertices to zero.

This gives a representation of theoriented matroid t. The oriented matroid y is represented by projecting the images

A COMBINATORIAL FORMULA FOR THE PONTRJAGIN CLASSES5of these vertices into (TxM ⊕R)/F n−1. By this construction, Y∆is decomposed intopieces indexed by elements of U∆.

One can see from stratified transversality theorythat if σ is generic, this decomposition can be refined to a Whitney stratificationY∆= Sα Sα that is transverse to the boundary. By construction, each stratum Sαdetermines an element u(Sα) ∈U∆by which piece of Y∆it lies in.A full flag of strata in a manifold is a set S = S0, S1, .

. .

, Sd where the closure ofSi contains Si−1, the dimension of Si is i, and d is the dimension of the manifold.If the manifold is oriented, the sign εS of S is defined as follows: Map a d-simplexwith vertices v0, . .

. , vd into the manifold so that the vertex v0 goes to S0, the edgev0v1 goes to S1, and so on.

Then εS = +1 if the orientation of the simplex agreeswith the orientation of the manifold, and εS = −1 otherwise. If S is a full flag ofthe strata in Y∆, then denote by u(S) the oriented simplex in C × Uδ with verticesu(S0), u(S1), .

. .

.For each simplex ∆of X, choose orientations [∆] of ∆and [D∆] of D∆whosecross product is the orientation of X. Orient Y∆by the cross product of [∆] andthe standard orientation of the Grassmannian of (n −1)-planes in (n + 1) space(remember that n is odd).Theorem 2. The generic smoothing σ induces a fixing cycle φ by the formulaφ =X∆XSε(S)c∆⋆(u(δ) × [D∆])where the first sum is over all simplices ∆of X and the second over all full flagsof strata in Y∆.The idea of the proof is to construct a continuous map f: Y →Y so that φ =f⋆[Y].5.

An alternative form of Chern-Weil theoryLet E be a vector bundle with a connection over a differentiable manifold M.Chern-Weil theory gives a formula for the Pontrjagin classes of M as a sum of terms,each of which is a product of curvature 2-forms Ωmultiplied by a pattern reflectingthe structure of the Lie algebra of Gl(n, R). Finding a combinatorial analogue ofΩis possible, but it is a singular current.

The difficulty in finding a combinatorialanalogue for Chern-Weil theory is regularizing the products.The combinatorial formula of this paper is an analogue of another form of Chern-Weil theory, which we now describe. Let e be the fiber dimension of E and assumethat it is even.

Let π: Y →M be the Grassmannian bundle of (e −2)-planes inE and ρ: Z →Y be the principle circle bundle of the tautological quotient 2-planebundle ξ over Y. The connection on E induces a one form Θ on Z with coefficientstwisted by the orientation sheaf O of Z and a curvature form Ωon Y defined byρ⋆Ω= dΘ.Proposition 4.

˜pi(E) = (−1)iπ⋆Ω(e−2+2i) where π⋆represents integration overthe fiber.Proposition 4 is proved by an algebraic manipulation using only the Whitneysum formula applied to π⋆E = ξ ⊕ξ⊥, the vanishing of high Pontrjagin classes of alow-dimensional bundle, and the projection formula. Theorem 1 is a combinatorialanalogue of this formula, where E is T M ⊕1.

6I. M. GELFAND AND R. D. MACPHERSONOrientations and dimensions.

Suppose that X is not orientable and/or not odddimensional. Let D be the orientation local system of X, so [X] ∈Hn(X, D).

Thefixing cycle should lie in homology with twisted coefficients: φ ∈Z3n−2(Y, π⋆D ⊗O⊗(n−1)). The construction of φ in §2 still works because Y has orientation sheafπ⋆D ⊗O⊗(n−1) where O is the orientation sheaf Z.References[BLSWZ] A. Bj¨orner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler, Oriented matroids,Encyclopedia Math.

Appl., Cambridge Univ. Press, 1992.[C]J.

Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18(1983), 575–657.[GGL]I.

Gabrielov, I. Gelfand, and M. Losik, Combinatorial calculation of characteristicclasses, Funktsional Anal. i Prilozhen.

9 (1975), 54–55; no. 2, 12–28; no.

3, 5–26.[M]R. MacPherson, The combinatorial formula of Garielov, Gelfand, and Loskik for thefirst Pontrjagin class, S´eminaire Bourbaki No.

497, Lecture Notes in Math. vol.

667Springer, Heidelburg, 1977.[N]N. Levitt, Grassmannians and gauss maps in piecewise-linear topology, Lecture Notesin Math., vol.

1366, Springer, Heidelburg, 1989.Department of Mathematics, Rutgers University, New Brunswick, New Jersey08903Department of Mathematics, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139

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