Differential characters and geometric invariants.

*(English)*Zbl 0621.57010
Geometry and topology, Proc. Spec. Year, College Park/Md. 1983/84, Lect. Notes Math. 1167, 50-80 (1985).

[For the entire collection see Zbl 0568.00014.]

From the summary: ”This paper first appeared in a collection of lecture notes which were distributed at the AMS Summer Institute of Differential Geometry, held at Stanford in 1973. We sketch the study of a functor which assigns to a smooth manifold M a certain graded ring \(\hat H^*(M)\), the ring of differential characters on M. Perhaps the main interest of our construction comes from the fact that the Weil homomorphism can be naturally factored through \(\hat H^*\). We should mention that our invariants are closely related to the differential forms TP(\(\theta)\) on the total space of a principal bundle with connection. These were considered by S. Chern and the second author [Ann. Math., II. Ser. 99, 48-69 (1974; Zbl 0283.53036)].

In fact, the present work arose out of the attempt to define objects in the base playing a role analogous to that of TP(\(\theta)\). Earlier results in this direction were formulated by the second author [”Characteristic forms and transgressions. II: Characters associated to a connection” (preprint)].”

From the summary: ”This paper first appeared in a collection of lecture notes which were distributed at the AMS Summer Institute of Differential Geometry, held at Stanford in 1973. We sketch the study of a functor which assigns to a smooth manifold M a certain graded ring \(\hat H^*(M)\), the ring of differential characters on M. Perhaps the main interest of our construction comes from the fact that the Weil homomorphism can be naturally factored through \(\hat H^*\). We should mention that our invariants are closely related to the differential forms TP(\(\theta)\) on the total space of a principal bundle with connection. These were considered by S. Chern and the second author [Ann. Math., II. Ser. 99, 48-69 (1974; Zbl 0283.53036)].

In fact, the present work arose out of the attempt to define objects in the base playing a role analogous to that of TP(\(\theta)\). Earlier results in this direction were formulated by the second author [”Characteristic forms and transgressions. II: Characters associated to a connection” (preprint)].”

Reviewer: P.Walczak

##### MSC:

57R20 | Characteristic classes and numbers in differential topology |

53C40 | Global submanifolds |

55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |

53C20 | Global Riemannian geometry, including pinching |

58A12 | de Rham theory in global analysis |

55R10 | Fiber bundles in algebraic topology |