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        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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        4
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        1answer
        241 views

        Using Stiefel-Whitney class to build new principal bundles

        I'm reading this paper and at the beginning of the second section, he states many results that aren't clear to me. Consider a principal $SO(3)$-bundle $P\rightarrow R^2\times \Sigma$, where $\Sigma$ ...
        5
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        1answer
        194 views

        A vector bundle associated to a codimension $1$ submanifold of a symplectic manifold

        We consider the standard symplectic structure $\omega=\sum dx_i\wedge dy_i$ on $\mathbb{R}^{2n}$. To every codimension $1$ submanifold $M\subset \mathbb{R}^{2n}$ we associate a vector bundle ...
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        1answer
        391 views

        Wu formula for manifolds with boundary

        The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...
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        0answers
        180 views

        Characteristic classes in a categorical framework [closed]

        How does the notion of Characteristic Classes behave in a Categorical framework? That is if instead of Manifolds and Smooth maps if we use Categories and Functors(Smooth in some way) and instead of ...
        3
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        0answers
        133 views

        Weil homomorphism

        In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to ...
        8
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        1answer
        164 views

        On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes

        In The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513 Woodward proposed a classification of $\mathrm{PU}...
        6
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        0answers
        77 views

        Relate two different mod 2 indices: $\eta$ invariant and the number of zero modes of Dirac operator, associated to SU(2)

        My major question in this post here is that: How can we relate the following two mod 2 indices: $\eta$ invariant, the number of the zero modes of the Dirac operator $N_0'$ mod 2, associated to ...
        1
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        0answers
        47 views

        Possibility of defining Chern character form in terms of odd Chern Character form?

        Given a complex vector bundle $E\to X$ with a connection $\nabla^E$ and an automorphism $U$ of $E\to X$, one can define an odd Chern character form $\textrm{ch}(\nabla, U)$ in terms of Chern character ...
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        0answers
        158 views

        Invariant polynomials in curvature tensor vs. characteristic classes

        Let $M$ be an $4m$-dimensional Riemannian manifold. We can then form the Pontryagin classes $p_k(TM)$ of the tangent bundle using Chern-Weil theory. For any sequence of numbers $k_1, \dots, k_l$ such ...
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        0answers
        167 views

        Generalizing the formula between Wu class and the Steenrod square

        I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy $$ Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) . \tag{eq.1}$$ ...
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        2answers
        353 views

        Is the cohomology ring $H^*(BG,\mathbb{Z})$ generated by Euler classes?

        I am interested in the classifying space $BG$ of a finite group $G$. A real representation $V$ of $G$ of dimension $r$ defines a real vector bundle over $BG$ of rank $r$. If the determinant of this ...
        1
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        0answers
        109 views

        Lift up characteristic class to chain complex

        In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...
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        190 views

        Generalize Wu formula to general Bockstein homomorphisms

        The classical Wu formula claims that $$Sq^1(x_{d-1})=w_1(TM)\cup x_{d-1}$$ on a $d$-manifold $M$, where $x_{d-1}\in H^{d-1}(M,\mathbb{Z}_2)$. I wonder whether there is a generalization of the ...
        5
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        1answer
        255 views

        Conversion formula between “generalized” Stiefel-Whitney class of real vector bundles: O(n) and SO(n)

        $O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$, $$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$ Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as: $$ w_j(...
        7
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        0answers
        223 views

        Different definitions of Stiefel-Whitney classes

        It is quite easy to show that different definitions of the Stiefel-Whitney classes agree by showing that they satisfy the well-known axioms. Nevertheless I have been asking myself wether one can prove ...

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