Riemann-Roch theorem for smooth manifolds
- Riemann-Roch theorem for smooth manifolds
In mathematics, a Riemann-Roch theorem for smooth manifolds is a version of results such as the Hirzebruch-Riemann-Roch theorem or Grothendieck-Riemann-Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure.
In their paper "Riemann-Roch theorems for differentiable manifolds" (Bull. Amer. Math. Soc. 65 (1959) 276-281) they define, for oriented smooth closed manifolds "X" and "Y" and a continuous mapping
:"f": "Y" → "X"
that "f" is a "c"1-map if there is "c"1 in the integral cohomology group
:"H"2("Y", "Z")
such that for the Stiefel-Whitney classes "w"2 we have
:"c"1 = "w"2("Y") − "f"*("w"2("X") modulo 2
in
:"H"2("Y", "Z"/2"Z").
Writing "ch"("X") for the image in "H"*("X", Q) they showed that for "f" a "c"1-map there is
:"f"!: "ch"("Y") → "ch"("X")
which is a homomorphism of abelian groups, and satisfying
:"f"!("y")"A"^("X") = "f"*("y".exp("c"1)/2)"A"^("Y")),
where "A"^ is the A-hat genus and "f"* the Gysin homomorphism. This mimics the GRR theorem; but "f"! has only an implicit definition.
This they specialised and refined in the case "X" = a point, where the condition becomes the existence of a spin structure on "Y". Corollaries are on Pontryagin classes and the J-homomorphism.
Wikimedia Foundation.
2010.
Look at other dictionaries:
Grothendieck–Hirzebruch–Riemann–Roch theorem — In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a… … Wikipedia
Atiyah–Singer index theorem — In the mathematics of manifolds and differential operators, the Atiyah–Singer index theorem states that for an elliptic differential operator on a compact manifold, the analytical index (closely related to the dimension of the space of solutions) … Wikipedia
Séminaire Nicolas Bourbaki (1950–1959) — Continuation of the Séminaire Nicolas Bourbaki programme, for the 1950s. 1950/51 series *33 Armand Borel, Sous groupes compacts maximaux des groupes de Lie, d après Cartan, Iwasawa et Mostow (maximal compact subgroups) *34 Henri Cartan, Espaces… … Wikipedia
List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… … Wikipedia
Gauss–Bonnet theorem — The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). It is named … Wikipedia
Plane curve — In mathematics, a plane curve is a curve in a Euclidean plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a … Wikipedia
Serre duality — In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non singular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a… … Wikipedia
Michael Atiyah — Sir Michael Atiyah Born 22 April 1929 (1929 04 22) (age 82) … Wikipedia
Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… … Wikipedia
Chern class — In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles. Chern classes were introduced by Shiing Shen Chern (1946). Contents 1 Basic… … Wikipedia
