Contents

Idea

In its most refined form, a secondary characteristic class is a characteristic class in ordinary differential cohomology. The term “secondary” refers to the fact that such a differential cohomology class in degree $n$ not only encodes a degree-$n$ class in integral cohomology, but in addition higher connection data in degree $(n+1)$: the data of a circle n-bundle with connection.

The refined Chern-Weil homomorphism takes values in such “secondary characteristic classes”.

But the precise meaning of the term secondary characteristic class varies a little in the literature, as follows. Historically it was first understood in more restricted senses.

1. In the strict sense of the word, a secondary characteristic class is a characteristic of a situation where an ordinary characteristic class vanishes (PetersonStein1962).

2. More specifically, a special case of this situation in differential geometry arises where the characteristic class is represented in de Rham cohomology by a curvature characteristic form. If that curvature form happens to vanish, the corresponding Chern-Simons form itself becomes closed, and now itself represents a cohomology class, in one degree lower. This is often called the corresponding Chern-Simons secondary characteristic class . Sometimes the term “secondary geometric invariants” is used for Chern-Simons forms (see for instance the review (FreedII)).

3. Using refined Chern-Weil theory the notions of curvature characteristic forms and their Chern-Simons forms are unified into the notion of cocycles in ordinary differential cohomology. The notion of Cheeger-Simons differential character was introduced to describe this unification, and it is has become tradition to call these differential characters themselves secondary characteristic classes independently of whether the corresponding ordinary characteristic class/curvature characteristic form vanishes or not (for instance (DupontKamber, Karlsson). More descriptively, this case is maybe better referred to as a differential characteristic class . See there for more details.

Related concepts

• Chern-Simons invariant

References

The notion in its general cohomological sense appears in

• Franklin P. Peterson, Norman Stein, Secondary characteristic classes , Annals of mathematics, Vol 76, No. 3 (1962)

The notion of Chern-Simons forms originates in

• Shiing-shen Chern, James Simons, Characteristic forms and geometric invariants The Annals of Mathematics, Second Series, Vol. 99, No. 1 (1974) (jstor).

The special meaning in the context of Chern-Weil theory in differential geometry was established by the introduction of Cheeger-Simons differential characters. Reviews of that include

• Dan Freed, Classical Chern-Simons theory II (pdf)

• Johan Dupont, Franz Kamber, Gerbes, Simplicial forms and Invariants for Families of Foliated Bundles (pdf)

• Michelle Karlsson, Characteristic classes and bounded cohomology (pdf)