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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Coefficient of the top Pontryagin class in $L$-genus

The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows: $$L_1=\frac{1}{3}p_1,$$ $$L_2=\frac{1}{45}(7p_2-p_1^2),$$ $$L_3=\frac{1}{945}(62p_3-...
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What is the Todd class *really*?

My question is about how to think about the Todd class. Usually this is presented via Grothendieck Riemann Roch (GRR): if $X$ is a smooth projective scheme over a field $\mathbf{C}$, the chern ...
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Stiefel Whitney number of a fiber bundle

I was going through this paper, and the author rights the following The Stiefel-Whitney class of $E$ is given by $$w(E)=(1+\alpha)^{2m+1}\left\{(1+c)^{2n+1}+u_1(1+c)^{2n}+\dots+u_{2n}(1+c)+u_{2n+1}\...
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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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Linear $S^{2k}$-bundles over $S^{4k}$

By the classification of Dold and Whitney, linear $S^2$-bundles over $S^4$ are classified by their first Pontryagin class $p_1$, which takes the value $4\lambda$ for the bundle corresponding to $\...
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Are there local maps of simplicial (co-)cycles on $d$-manifolds beyond cohomology operations?

I'm interested in locally defined maps of cocycles/chains on manifolds of a fixed dimension $d$ which are compatible with cohomology. To be concrete about what "local" means, let me consider ...
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Preimage by birational maps

I am looking for an example (I guess that in complex projective space $\mathbb{P}^{n}$ is good) such that satisfy the following condition (in non trivial case, for this assume $X \neq \tilde{X}$): Let ...
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Multi-variable cohomology operations

Intuitively, cohomology operations are ways to locally compute a cocycle $\alpha\in H^i(X, G)$ from any cocycle $\beta\in H^j(X, H)$. Formally, they are in one-to-one correspondence with homotopy ...
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What characteristic classes are there?

Can someone concisely list all characteristic classes (i.e., the cohomology classes $H^*(BX,A)$ of the corresponding classifying spaces) for the most relevant structure groups $X$ such as $O(n)$, $SO(...
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Do we know any examples of complex surfaces where we have explicit knowledge of the Chern–Weil functions?

Let $X$ be a compact complex surface (smooth). Let $\gamma_1, \gamma_2$ denote the Chern–Weil functions. That is, if $\omega$ is a Kähler form on $X$ with volume form $\omega^2$, then $\gamma_1, \...
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Define a characteristic class on a simplicial complex (non-manifold)

Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class? (Please provide Yes or No answers, and reasonings.) Given a fixed ...
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Calculation about Chern character in a special setting

I'm confused with working out the Chern character in the following special setting. Let $E$ be a spinor bundle $$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$ over sphere $S^{2n}$, where $\rho$ ...
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Real analogue of Segre classes

Let $X$ be a manifold and $E\to X$ a complex vector bundle and let's work in $H^\bullet(X,\mathbb{Z})$. Given the total Chern class of $E$, $c(E)=1+c_1(E)+\cdots+c_n(E)$, we can define the total Segre ...
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Rank of matrix coming from cobordism computations

In a computation of Pontryagin-numbers of certain manifolds (see the appendix of https://arxiv.org/pdf/2109.10306.pdf for more context) we came across the following elementary problem: Consider the ...
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Fibering cobordant to projectivization of a vector bundle

I was going through this Stong's paper, I am stuck in the proof of the proposition 8.4 (given below) I understand the proof till he derives the expression for the Steenrod square operation of the ...
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How much do characteristic classes fail to characterize bundles?

Given a group $G$, let $E \to B$ be a principal $G$-bundle. It is well-known that when $B$ is a nice enough topological space (e.g. CW-complex), such a thing corresponds to a connected component of $...
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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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7 votes
2 answers
376 views

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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3 answers
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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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2 votes
1 answer
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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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2 votes
1 answer
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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 ...
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Push forward of Chern character and index theorem

I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998). I expose here the setup for my ...
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How to calculate the total chern classes of CP^n [closed]

When calculating the total chern class of $\ CP^n$, we use the fact that their is a exact sequence of vector bundles over $\ CP^n$: $$\ 0\to S \to C^{n+1} \to Q \to 0$$ And identify the bundle $\ TCP^...
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Visualize how the 5d Dold manifold and Wu manifold are cobordant via a 6d manifolds with boundaries

Are there simple intuitions and arguments to visualize why the following two 5-manifolds are cobordant to each other with the oriented structures? (They can be two boundaries of 6-dimensional oriented ...
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3 votes
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The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$

$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient, it is often to see that we say the 1st Stiefel Whitney class $$...
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Evaluating the Euler class of a circle bundle on fibers

I am trying to understand what kind on information the Euler class provides about certain submanifolds of a given circle bundle. This might be completely obvious, but I don't see how to answer the ...
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4 votes
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Characteristic classes of quotient manifold

Let $M$ be a compact oriented smooth manifold with boundary and let $G$ be a compact Lie group acting smoothly, orientation-preservingly and freely on $M$. (Under what conditions) is there a ...
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9 votes
2 answers
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Differential refinement of homology

Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism ...
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2 votes
1 answer
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Calculating topological $K(X)$ for complex projective manifolds

In the introduction to the book Vector bundles and K-theory http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html two approaches to classification of (topological) vector bundles are discussed - the ...
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10 votes
1 answer
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Chern classes of a mapping torus vector bundle in terms of the construction data

Let $\pi:E\to X$ be a complex vector bundle*, and $f:E\to E$ a bundle isomorphism. Consider the mapping torus $$E(f) := \frac{E\times [0,1]}{E \times \{0\}\sim_f E \times \{1\}}$$ where the ...
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Relation between Bott-Chern forms and Second fundamental form

Given a short exact sequence of holomorphic Hermitian vector bundles $$0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0,$$ the second fundamental form measures the obstruction of $E\simeq F\oplus ...
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Which classes in $\mathrm{H}^4(B\mathrm{Exceptional}; \mathbb{Z})$ are classical characteristic classes?

Let $G$ be a compact connected Lie group. Recall that $\mathrm{H}^4(\mathrm{B}G;\mathbb{Z})$ is then a free abelian group of finite rank. Let us say that a class $c \in \mathrm{H}^4(\mathrm{B}G;\...
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5 votes
1 answer
138 views

Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$

For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e., $$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(...
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4 votes
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Instanton numbers for diverse gauge bundles on diverse manifolds --- their relations to characteristic classes

It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n \in \mathbb{Z}$ $$ n = { 1 \over 8\pi^2} \int_{\mathcal{M}_{4}} \text{tr} \left(F \wedge F\right) = {1 \...
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12 votes
1 answer
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Realizing Stiefel-Whitney classes via vector bundles

Let $X$ be a CW complex. If $E$ is a vector bundle over $X$, then it's well-known that the Stiefel-Whitney classes $w_j(E) \in H^j(X,\mathbb F_2)$ of $E$ are determined from the classes $w_{2^k}(E)$ (...
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7 votes
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Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?

This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case. Let $M$ be a ...
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