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        5
        votes
        0answers
        169 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 $$ Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) . \tag{eq.1}$$ ...
        4
        votes
        0answers
        219 views

        Generalized Postnikov square

        Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...
        3
        votes
        1answer
        122 views

        Bockstein homomorphism and Square Operations: Their consistency formulas

        Here are various ways to define "Bockstein homomorphism:" Let $\beta_p:H^*(-,\mathbb{Z}_p) \to H^{*+1}(-,\mathbb{Z}_p)$ be the Bockstein homomorphism associated to the extension $$\mathbb{Z}_p\to\...
        2
        votes
        0answers
        107 views

        Pontryagin square on spin and non-spin manifold

        The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
        2
        votes
        0answers
        81 views

        Do we have a transgression formula for the chern characters of quasi-isomorphic cochain complexes of vector bundles?

        Let $(E^{\cdot},d_E^{\cdot})$ be a cochain complex of complex vector bundles on a smooth compact manifold $X$. Now for each $E^i$ we could assign a connection $\nabla_E^i$ and obtain its curvature $(\...
        3
        votes
        0answers
        141 views

        Cohomology classes functorial under etale morphisms

        Consider the full subcategory $\mathcal C$ in the big etale site of a field $k$ consisting of smooth schemes, namely the category of smooth varieties over a field $k$ with etale maps as morphisms. I ...
        13
        votes
        0answers
        724 views

        Local proof of Grothendieck-Riemann-Roch theorem

        There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem. Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...

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