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Questions tagged [cohomological-operations]

Cohomological operations are morphisms between cohomology groups of a space, natural with respect to the space. Most common examples include Steenrod squares in mod 2 cohomology and Adams operations in K-theory.

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Fiber integration is independent of the operations involved.

In Definition 2 of Fiber Integration nlab post, the author claimed that the operation is indpendent of the choices involved. How is this so? The post itself is quite long. I think it is easier ...
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Why is the Bockstein morphism a derivation?

I'm trying to understand the Bockstein morphism in cohomology, and one of the points is that $\delta : H^*(G,\mathbb{F}_p)\to H^*(G,\mathbb{F}_p)$ is a derivation that squares to $0$. I could ...
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Order relation between cohomology groups.

We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex $$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...
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Exact sequence of cohomology $0 \rightarrow H^{n - 1}(X) \rightarrow H^{n - 1}(Y) \rightarrow \mathbb{Z}^2 \rightarrow \mathbb{Z} \rightarrow 0$

Say I have the following exact sequence of cohomology: \begin{align*} 0 \rightarrow H^{n - 1}(X) \rightarrow H^{n - 1}(Y) \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow 0 \...
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Importance of the Cup Product in Topology?

On p. 15 of Fenn's Techniques of Geometric Topology, it says: In many ways, the cup product [for cohomology] is the 'soul' inside all topological existence theorems. Statements like: two lines in ...
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Reference for secondary cohomology operations

I am learning some homotopy theory and am currently reading Mosher and Tangora. I love the content of this book, it's very terse and comes straight to the point. At the same time I find it very ...
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The smallest $n> 0$ with the nonzero $n$th Stiefel-Whitney class is a power of 2 when total Stiefel-Whitney class is not trivial.

This is the Problem 8-B form the characteristic classes by John W. Milnor and James D. Stasheff. [Problem 8-B]. If the total Stiefel-Whitney class $w(\xi) \neq 1$, show that the smallest $n>0$ ...
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Calculate the Wu class from the Stiefel-Whitney class

The total Stiefel-Whitney class $w=1+w_1+w_2+\cdots$ is related to the total Wu class $u=1+u_1+u_2+\cdots$: The total Stiefel-Whitney class $w$ is the Steenrod square of the Wu class $u$: \begin{align}...
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differential in AHSS for spin cobordism

According to these solutions, the differential $d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})$ is the dual of $Sq^2$. Why? This MO post asks a similar question (but about $d_3$ in ...
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Steenrod Algebra as automorphisms of additive group

Is there a direct way to see that the subalgebra of the mod-$p$ Steenrod algebra ${\mathcal A}_p$ generated by the reduced powers is isomorphic to the dual of the Hopf algebra ${\mathcal O}(\text{Aut}(...
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Why does «Massey cube» of an odd element lie in 3-torsion?

The cup product is supercommutative, i.e the supercommutator $[-,-]$ is trivial at the cohomology level — but not at the cochain level, which allows one to produce various cohomology operations. The ...
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The simplest nontrivial (unstable) integral cohomology operation

By an integral cohomology operation I mean a natural transformation $H^i(X, \mathbb{Z}) \times H^j(X, \mathbb{Z}) \times ... \to H^k(X, \mathbb{Z})$, where we restrict $X$ to some nice category of ...
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Algebra homomorphism in Steenrod algebra.

Let $\psi$ be the map of generators $\psi : \mathfrak{a}(2) \rightarrow \mathfrak{a}(2) \otimes \mathfrak{a}(2)$. $$\psi(Sq^k) = \sum_{i=0}^k Sq^i \otimes Sq^{k-i}$$ Let $\underline{\mathfrak{a}}$ be ...
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“Since $H^*(X)$ is a polynomial ring, $H^{q + 2^i} =0$ for $0 < 2^i <q$”?

I have some questions in the proof: "Theorem: If $H^*(X,\mathbb{Z}_2)$ is a polynomial ring or a truncated polynomial ring in a generator $x$ of dimension $q$, and $x^2 \neq 0$, then $q=2^k$ for some ...
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Prove that $s \operatorname{Sq}^i = \operatorname{Sq}^i s$.

I have been studying the mod 2 Steenrod algebra. And I try to solve some exercises of it. Can you help me to check this proof: Let $SX$ denote the suspension of $X$, and let $S: \underline{H}^q(X) ...
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$f^\ast (a \smile b) = f^\ast(a) \smile f^\ast(b)$ using simplicial chains to define cochains

Let $f \colon X \to Y$ be a continuous map between topological spaces $X$ and $Y$, $f_\ast$ be the induced homomorphism of singular chains $C_k^s(X;G)$, $C_k^s(Y;G)$ and $f^\ast$ be the induced ...
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The “Wu formula” and Steenrod algebras

The Wikipedia page on Stiefel-Whitney classes includes the following paragraph: Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth manifold (defined as the Stiefel–Whitney classes ...
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Steenrod squares on the sphere

Let $S^d$ denote the $d$-sphere. The only non-trivial cohomology groups are $H^0(S^d;\mathbb Z_2)= \mathbb Z_2$ generated by $1$ and $H^d(S^d;\mathbb Z_2)= \mathbb Z_2$ generated by the fundamental ...
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Steenrod Algebra - Converting between Milnor to Serre-Cartan basis'

I have been studying the mod 2 Steenrod Algebra (denoted $\mathcal{A}$), using Mosher & Tangora. We have the Serre-Cartan (or Adem basis): Let $I = \{i_1,i_2,\ldots,i_n\}$ be a sequence of ...
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The action of the Steenrod algebra on $H^*(BU; \mathbb{Z}_p)$

By considering the classifying map $f \colon (\mathbb{C} P^{\infty})^n \rightarrow BU(n)$, its induced map on cohomology, and using the Cartan formula, we can derive the Wu formula for the action of ...