Israel M. Gelfand, Robert D. MacPherson
A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given. The main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.
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In 1992 Gelfand and MacPherson gave a local and combinatorial formula for the Pontrjagin classes of a differential manifold. We give an expanded version of their discussion and highlight the origins of combinatorial differential manifolds in their work.
The theory of indices of Morse--Bott vector fields on a manifold is constructed and the famous localization problem for the transfer map is solved on its base in the present paper. As a consequence, we obtained addition theorems for the universal Pontrjagin characteristic classes in cobordisms. These results gave us a possibility to complete the construction, which was begun more than twenty years ago, of the universal characteristic classes' theory.
Local formulae for combinatorial Pontrjagin classes
本文证明组合流形的庞特里亚金类多项式示性类可由仅依赖单形链环组合类型的局部有理单纯链表示，并刻画了第一庞特里亚金类的全部此类局部公式及其系数分母估计。
A Bott manifold is the total space of some iterated $\mathbb C P^1$-bundle over a point. We prove that any graded ring isomorphism between the cohomology rings of two Bott manifolds preserves their Pontrjagin classes. Moreover, we prove that such an isomorphism is induced from a diffeomorphism if the Bott manifolds are $\mathbb Z/2$-trivial, where a Bott manifold is called $\mathbb Z/2$-trivial if its cohomology ring with $\mathbb Z/2$-coefficient is isomorphic to that of a product of $\mathbb C P^1$'s.
We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any compact Lie group G. In the special case when G is finite or a torus, we show that our construction recovers the construction of Wasserman, providing a new perspective on equivariant bordism. We connect the results with bordisms of global quotient orbifolds, utilizing the machinery of Gepner-Henriques to describe bordisms of framed orbifolds in terms of equivaria…
A torus manifold is a closed smooth manifold of dimension $2n$ having an effective smooth $T^n = (S^1)^n$-action with non-empty fixed points. Petrie \cite{petrie:1973} has shown that any homotopy equivalence between a complex projective space $\CP^n$ and a torus manifold homotopy equivalent to $\CP^n$ preserves their Pontrjagin classes. A \emph{generalized Bott manifold} is a closed smooth manifold obtained as the total space of an iterated complex projective space bundles over a point, where each fibration is a projectivization of the Whitney sum of a finite many complex line bundles. For ins…
This paper studies the interaction of $π_1(M)$ for a $C^\infty$ manifold $M$ with Bott's original obstruction to integrability, and with differential geometric invariants such as Godbillon-Vey and Cheeger-Simons invariants of a foliation. We prove that the ring of higher Pontrjagin and higher Chern classes of an integrable subbundle $E$ of the tangent bundle of a manifold vanishes above dimension $2k$ where $k=dim(TM/E)$, and where the higher Pontrjagin and Chern rings are rings generated by $i^*y \cup p_j(TM/E)$ and by $i^*y \cup c_j(TM/E)$ respectively, with $p_j$ the $j$-th Pontrjagin class…
We give a combinatorial construction of an ordered semiring A, and show that it can be identified with a certain subquotient of the semiring of p-local Bousfield classes, containing almost all of the classes that have previously been named and studied. This is a convenient way to encapsulate most of the known results about Bousfield classes.
The combinatorial description via ribbon graphs of the moduli space of Riemann surfaces makes it possible to define combinatorial cycles in a natural way. Witten and Kontsevich first conjectured that these classes are polynomials in the tautological classes. We answer affirmatively to this conjecture and find recursively all the polynomials.
Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all powers of first Chern class in the terms of mathematical expectations of parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of triangulated circle bundle over simplex, measuring mixing by triangulation of the circular graphs over …
Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid bundles to spherical quasifibrations. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. This shows the mod 2 cohomology of the poset of rank k oriented matroids…
The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem. We solve this and incorporate the results into properties of the real moduli space of Riemann spheres. Connections are discussed with relation to classic combinatorial problems as well as to other sciences.
Cohomological rigidity of manifolds with torus actions: I
本文针对 moment-angle 流形与拓扑环面流形，证明满足组合条件的单纯球面所对应流形具有上同调刚性，并通过面环的 Tor-代数确定其组合类型。
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