# Questions tagged [characteristic-classes]

Cohomology classes associated to vector bundles. Includes Stiefel-Whitney classes, Chern classes, Pontryagin classes, and the Euler class.

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### First Chern form of line subbundle

Let $\pi:E\to X$ be a holomorphic vector bundle over a complex manifold. Denote by $\tilde{E}=\pi^*E\to E$ the pullback of $E$ over itself. There exists a tautological line bundle $L\subset \tilde{E}$ ...
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### First Pontrjagin class and generator of $\pi_3(\mathrm{SO}(d))$

It is well-known that $H^4(B\mathrm{SO}(d), \mathbb{Z}) \cong \mathbb{Z}$, with a canonical generator given by $p_1$, the first universal Pontrjagin class. Let's assume $d\geq 5$ so that everything is ...
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### When are bundles of odd and even differential forms isomorphic?

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$odd}} \Omega^k$ ...
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### Cobordism class of projectivization of a bundle

I was reading the book "Differentiable Periodic Maps" by P.E. Conner (1979). I am stuck at the following problem given at the end of section 21: Let $\xi\to V^n$ be a $k$-plane bundle over a ...
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### Which Stiefel-Whitney numbers can be extended to manifolds with boundaries?

The Stiefel-Whitney numbers are classical topological manifold invariants obtained by integrating some local quantity (a cup product of Stiefel-Whitney classes) over the manifold. Which Stiefel-...
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### Justification for the definition of equivariant curvature

Let $G$ be a compact Lie group which act on a smooth manifold $M$. Let $\mathbb{C}[\mathfrak{g}] \otimes \mathcal{A}$ be the algebra of polynomial maps from $\mathfrak{g}$ to $\mathcal{A}(M),$ we ...
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### Chern-Weil theory on some noncompact groups, and characteristic classes in differential cohomology

$\newcommand{\Z}{\mathbb Z}\newcommand{\HdR}{H_{\mathrm{dR}}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\g}{\mathfrak g}$I have a specific question about invariant polynomials for some Lie groups, ...
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### LaTeXed "Lectures on characteristic classes" [closed]

I don't know if this is the right place to ask, but... Is anyone interested in a LaTeXed version (by me) of "Lectures on characteristic classes" by Milnor in 1957? Of course the successor &...
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### Different ways of defining the Chern character of a complex

Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form $$0 \to E_N \to E_{N-1} \to \dots \to {E_0} \to 0,$$ where the bundles are ...
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### Discrete spectrum of Dirac operator

It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete. For example at least for $d=4$, this ...
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### Even, non liftable Stiefel-Whitney class

Let $M$ be a smooth manifold and $E$ a smooth real vector bundle of even rank over $M$. If $E$ admits of a complex vector bundle structure $\mathcal E$ ($\mathcal E_\mathbb R=E$) then all odd Stiefel-...
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### Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?

The Whitney sum formula for Stiefel-Whitney classes, $w_n(V \oplus W) = \sum w_i(V) w_{n-i}(W)$, looks a lot like the one for Chern classes $c_n(V \oplus W) = \sum c_i(V)c_{n-i}(W)$. But I don't know ...
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### Comparing the Segre classes of a cone with its abelian hull

Let $X$ be a smooth scheme, with a sheaf of graded quasi-coherent algebras $\mathcal{A}^*$, that yields a cone $C$ (in the sense of Fulton's intersection theory). Suppose that $\mathcal{A}^1$ is a ...
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### Projectively flat connection

Let $E \to B$ be a Hermitian vector bundle. If $E$ has a projectively flat connection, then its total Chern character has the form $\mbox{ch}(E) = \mbox{rank} \cdot \exp(\mbox{slope})$. Is the ...
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### Whitney sum via Gysin

Let $E_1\to E\to E_2$ be a short exact sequence of vector bundles. The Whitney sum formula says that $e(E)=e(E_1)e(E_2)$, i.e. that the Euler class is multiplicative. Is there a proof of this fact ...
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### Motivation for the definition of complex orientable cohomology theory

PRELIMINARY DEFINITIONS: Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have: $$\tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt)$$ So there is a special ...
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### Curvature as infinitesimal holonomy 2

This question may be seen as a follow up of this original question. I'm learning Cheeger-Simons differential characters (reading Differential Characters of Bär and Becker). If I understand correctly, ...
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### de Rham-invariants of a Riemannian metric

$\DeclareMathOperator{\Sym}{Sym}$For $N>0$, consider the $O_N$-representations $V = \mathbb R^N$ and $M_n = \ker (\Sym^n{V}\otimes\Sym^2 V\to \Sym^{n+1} V\otimes V)$ (the irreducible $GL_n$-...
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### Shulman's Thesis on Characteristic Classes

I am trying to find a copy of H. Shulman's 1972 Berkeley thesis 'On Characteristic Classes'. I've seen it referenced in Bott's 'On the de Rham theory of Certain Classifying Spaces' but I can't seem to ...
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### The maximum number of vertical independent vector fields on the tangent bundle

Let $M$ be a differentiable manifold. Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$? Is there a name for ...