algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
By the Hirzebruch-Riemann-Roch theorem the index of the Dolbeault operator is the Todd genus (e.g. Gilkey 95, section 5.2 (more generally so for the Spin^c Dirac operator).
The characteristic series of the Todd genus is
This means that as a formal power series in Chern classes the Todd class starts out as
On an almost complex manifold , the Todd class coincides with the A-hat class up to the exponential of half the first Chern class:
(e.g. Freed 87 (1.1.14)).
In particular, on manifolds with SU-structure, where , the Todd class is actually equal to the A-hat class:
(rational Todd class is Chern character of Thom class)
Let be a complex vector bundle over a compact topological space. Then the Todd class of in rational cohomology equals the Chern character of the Thom class in the complex topological K-theory of the Thom space , when both are compared via the Thom isomorphisms :
More generally , for any class, we have
which specializes to the previous statement for .
(Karoubi 78, Chapter V, Theorem 4.4)
We discuss how the e-invariant in its Q/Z-incarnation (this Def.) has a natural formulation in cobordism theory (Conner-Floyd 66), by evaluating Todd classes on cobounding (U,fr)-manifolds.
This is Prop. below; but first to recall some background:
In generalization to how the U-bordism ring is represented by homotopy classes of maps into the Thom spectrum MU, so the (U,fr)-bordism ring is represented by maps into the quotient spaces (for the canonical inclusion):
The bordism rings for MU, MUFr and MFr sit in a short exact sequence of the form
where is the evident inclusion, while is restriction to the boundary.
In particular, this means that is surjective, hence that every -manifold is the boundary of a (U,fr)-manifold.
(e-invariant is Todd class of cobounding (U,fr)-manifold)
Evaluation of the Todd class on (U,fr)-manifolds yields rational numbers which are integers on actual -manifolds. It follows with the short exact sequence (2) that assigning to -manifolds the Todd class of any of their cobounding -manifolds yields a well-defined element in Q/Z.
Under the Pontrjagin-Thom isomorphism between the framed bordism ring and the stable homotopy group of spheres , this assignment coincides with the Adams e-invariant in its Q/Z-incarnation:
(Conner-Floyd 66, Theorem 16.2)
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
Named after John Arthur Todd.
Original articles:
Friedrich Hirzebruch, Section 3 of: Neue topologische Methoden in der Algebraischen Geometrie, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 (doi:10.1007/978-3-662-41083-7)
English translation: Section 3 of Topological Methods in Algebraic Topology, Classics in Mathematics, vol 131. Springer 1995 (doi:10.1007/978-3-642-62018-8_4)
Pierre Conner, Edwin Floyd, Sections 12, 13 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)
Max Karoubi, Chapter V.4 of: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 (pdf, doi:10.1007%2F978-3-540-79890-3)
Review:
On the Todd character:
Pierre Conner, Larry Smith, Section 7 of: On the complex bordism of finite complexes. II, J. Differential Geom. Volume 6, Number 2 (1971), 135-174 (euclid:euclid.ijm/1256051760)
Larry Smith, Section 1 of: The Todd character and the integrality theorem for the Chern character, Illinois J. Math. Volume 17, Issue 2 (1973), 301-310 (euclid:ijm/1256051760)
Larry Smith, The Todd character and cohomology operations, Advances in Mathematics Volume 11, Issue 1, August 1973, Pages 72-92 (doi:10.1016/0001-8708(73)90003-0)
Review:
See also:
Discussion of Todd classes for noncommutative topology/in KK-theory is in
Review of the Todd genus with an eye towards generalization to the Witten genus is in the introduction of
Last revised on February 22, 2021 at 12:39:27. See the history of this page for a list of all contributions to it.