algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $\mathcal{V}_X$ a complex vector bundle of complex rank $n$, the highest degree Chern class that may generally be non-vanishing is $c_n$. This is hence often called the top Chern class of the vector bundle.
The top Chern class of a complex vector bundle $\mathcal{V}_X$ equals the Euler class $e$ of the underlying real vector bundle $\mathcal{V}^{\mathbb{R}}_X$:
(e.g. Bott-Tu 82 (20.10.6))
The top Chern class of a complex vector bundle $\mathcal{V}_X$ equals the pullback of any Thom class $th \;\in\; H^{2n}\big( \mathcal{V}_X; \mathbb{Z} \big)$ on $\mathcal{V}_X$ along the zero-section:
(e.g. Bott-Tu 82, Prop. 12.4)
This relation to Thom classes generalizes to Conner-Floyd-Chern classes in complex oriented Whitehead-generalized cohomology. See at universal complex orientation on MU.
For more references see at Chern class and at characteristic class.
Created on January 26, 2021 at 08:52:54. See the history of this page for a list of all contributions to it.