Superconnections, Thom classes, and equivariant differential forms.

*(English)*Zbl 0592.55015Let E be a vector bundle of even rank over a manifold M, E equipped with a spin structure, \(i: M\to E\) be the zero section and \(i_ !1\) be the Thom class in K-theory which, according to the considerations from the paper by M. F. Atiyah, R. Bott and A. Shapiro [ibid. 3, Suppl. 1, 3-38 (1964; Zbl 0146.190)] is explicitly represented by a super vector bundle with odd endomorphism constructed from the bundle of spinors associated to the spin structure.

The computation of the character form of this representation for \(i_ !1\), by applying the superconnection formalism from the paper by D. Quillen [ibid. 24, 89-95 (1985; Zbl 0569.58030)] is the principal aim of this paper. Moreover, the Thom class in cohomology \(i_*l\) is represented by a differential form having a nice Gaussian shape peaked along the zero section.

Since this Thom form is obtained by a complicated method involving in an essential way the spin structure and the Clifford algebra calculations, the authors also give an independent and more direct construction of this Gaussian shaped Thom form based on the Pfaffian algebra. For this purpose, by using the machinery of the Weil algebra and equivariant forms, the authors achieve a universal algebraic situation where the curvature matrix can be assumed invertible. The Gaussian Thom form is used to give a simple approach to transgression in the sphere bundle.

The computation of the character form of this representation for \(i_ !1\), by applying the superconnection formalism from the paper by D. Quillen [ibid. 24, 89-95 (1985; Zbl 0569.58030)] is the principal aim of this paper. Moreover, the Thom class in cohomology \(i_*l\) is represented by a differential form having a nice Gaussian shape peaked along the zero section.

Since this Thom form is obtained by a complicated method involving in an essential way the spin structure and the Clifford algebra calculations, the authors also give an independent and more direct construction of this Gaussian shaped Thom form based on the Pfaffian algebra. For this purpose, by using the machinery of the Weil algebra and equivariant forms, the authors achieve a universal algebraic situation where the curvature matrix can be assumed invertible. The Gaussian Thom form is used to give a simple approach to transgression in the sphere bundle.

Reviewer: L.Maxim

##### MSC:

55R50 | Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory |

55T10 | Serre spectral sequences |

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

55R25 | Sphere bundles and vector bundles in algebraic topology |

58A12 | de Rham theory in global analysis |