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        Questions tagged [cohomology]

        A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...

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        Cohomological indices VS cup-length, Lusternik-Schnirelmann category, betti sum in critical point set theory

        Let $M$ be a (closed) manifold acted on by a compact Lie group $G$. For any characteristic class $\alpha \in H^*(BG)$, one can define a so-called cohomological index of $M$ as follows: $$\text{ind}_{\...
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        132 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 ...
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        Alternating property of H_2(T, Z)

        Let us consider the torus $T = S^1_X \times S^1_Y$, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. We have an alternate property $dX \wedge dY = - dY \wedge dX \...
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        0answers
        72 views

        A relation between Hochschild cohomology of a $C^*$ algebra and its bidual

        Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product. Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$...
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        1answer
        142 views

        Alternate property of H^2(T, Z) [closed]

        Let us take $T = S^1_X \times S^1_Y$, which is a torus, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. If we consider the generator $dX \wedge dY \in H^2(T, {\Bbb ...
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        1answer
        163 views

        The sheafification of taking cohomology is trivial?

        Consider the Nisnevich site of a noetherian scheme $S$ of finite Krull dimension (the objects are schemes $U$ smooth and of finite type over $S$), let $A$ be a sheaf of abelian groups on this site. I ...
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        1k views

        What is Prismatic cohomology?

        Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...
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        1answer
        73 views

        Gysin morphism of blow up

        Let $X$ be a smooth, projective variety and $i:Y \hookrightarrow X$ a smooth divisor. Let $Z \subset X$ be a proper, closed subvariety disjoint from $Y$. Let $\pi:\widetilde{X} \to X$ be the blow-up ...
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        votes
        1answer
        160 views

        Invariants in relative cohomology and compact support cohomology of the quotient

        Let $\cal H$ be the Poincare upper half-plane and $\overline {\cal H}$ the union of $\cal H$ with the set of cusps $\bf P^1 (\bf Q)$, provided with its usual topology. Let $\Gamma$ a congruence ...
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        1answer
        249 views

        Topology of connected subsets of the $3$-torus

        Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$. We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded. I am ...
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        0answers
        86 views

        Is the following variant of the Universal Coefficient Theorem valid?

        A version of the Universal Coefficient Theorem that relates the integer cohomology of a group $G$ to its cohomology with coefficients in an abelian group $M$ is as follows: $H^n(G,M) = H^n(G,\mathbb ...
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        vote
        1answer
        140 views

        About Hom and weight space of nilpotent Lie algebra cohomology

        Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $\Phi$ the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ ...
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        votes
        1answer
        85 views

        Convergence of the Lyndon-Hochschild-Serre spectral sequence as an algebra

        Consider a short (not necessarily split) exact sequence of groups $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ and suppose we wish to find the cohomology of $G$ with coefficients in a ...
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        1answer
        365 views

        Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology?

        There are numerous expositions of simplicial homology in the literature. Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes. Hatcher in “Algebraic ...
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        2answers
        448 views

        What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian?

        Let $\operatorname{Gr}(k, n)$ and $\operatorname{Gr}^+(k, n)$ denote the unoriented and oriented grassmannians respectively. The $\mathbb{Z}_2$ cohomology of the unoriented grassmannian is $$H^*(\...

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