Characteristic class

A characteristic class is a mathematical object from the differential topology . It is a topological invariant of a vector bundle and can be represented by a differential form. A characteristic class describes more or less the “twistedness” of a bundle, so the characteristic class of a trivial bundle mostly corresponds to the one element.

definition

Be or . If a vector bundle with fiber and the Graßmann manifold , then a mapping that is unambiguous except for homotopy can be defined, which is superimposed on the tautological bundle by a bundle mapping . ${\ displaystyle k = \ mathbb {R}}$ ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle \ pi \ colon E \ to X}$ ${\ displaystyle V \ simeq k ^ {n}}$ ${\ displaystyle BG}$ ${\ displaystyle G_ {n} (k ^ {\ infty})}$ ${\ displaystyle f \ colon X \ to BG}$ ${\ displaystyle F \ colon E \ to \ gamma ^ {n}}$ ${\ displaystyle BG}$

Be a commutative ring with one element. For each cohomology class the characteristic class is defined by ${\ displaystyle R}$ ${\ displaystyle c \ in H ^ {*} (BG; R)}$ ${\ displaystyle c (E)}$

{\ displaystyle c (E): = f ^ {*} (c) \ in H ^ {*} (X; R) \ ,.}

motivation

An n-dimensional vector bundle is trivial if and only if its classifying mapping is null homotop (homotop to a constant mapping). However, this condition is difficult to check. It is easier to check whether the induced mappings are trivial in homology or cohomology and this is exactly what is measured by characteristic classes. ${\ displaystyle \ pi \ colon E \ to X}$ ${\ displaystyle f \ colon X \ to BG}$

Examples

Boot-Whitney classes of real vector bundles
Euler class of oriented real vector bundles
Chern classes of complex vector bundles
Pontryagin classes of real vector bundles

Principal Bundle

More generally, one can define characteristic classes of principal bundles . Each cohomology class of the classifying space of the Lie group corresponds to a characteristic class of principle bundles . This is defined by , where the classifying map is from . ${\ displaystyle c \ in H ^ {*} (BG)}$ ${\ displaystyle BG}$ ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle \ pi: P \ rightarrow B}$ ${\ displaystyle c (P): = f ^ {*} (c) \ in H ^ {*} (B)}$ ${\ displaystyle f: B \ rightarrow BG}$ ${\ displaystyle \ pi}$

In the case of or , the characteristic classes of principal bundles correspond to the characteristic classes of the associated vector bundles . ${\ displaystyle G = GL (n, \ mathbb {R})}$ ${\ displaystyle G = GL (n, \ mathbb {C})}$ ${\ displaystyle G}$

Conversely, for each (real or complex) vector bundle provided with a metric, the frame bundle can be regarded as a principal bundle (with a structure group or ) whose characteristic classes correspond to the characteristic classes of the vector bundle. ${\ displaystyle G = GL (n, \ mathbb {R})}$ ${\ displaystyle G = GL (n, \ mathbb {C})}$

Characteristic classes of principal bundles can be calculated using the Chern-Weil theory from the curvature of a connection . In particular, the characteristic classes of flat bundles disappear. For these one can then define secondary characteristic classes .

literature

Edwin H. Spanier: Algebraic Topology. 1. corrected Springer edition, reprint. Springer, Berlin et al. 1995, ISBN 3-540-90646-0 .
Allen Hatcher: Vector Bundles and K-Theory (PDF; 1.2 MB)
May: A concise course in algebraic topology (Chapter 23: "Characteristic classes of vector bundles")