Questions tagged [cohomology-operations]
Use this tag for questions about natural transformations from a functor defining a cohomology theory to itself. Common examples include Steenrod squares in mod 2 cohomology and Adams operations in K-theory.
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Steenrod squares in a Puppe sequence
Consider the following Puppe sequence.
$$\cdots\to K(\mathbb{Z}/2\mathbb{Z},2)\to \Omega X\to K(\mathbb{Z}/2\mathbb{Z},1)\stackrel{a}{\to} K(\mathbb{Z}/2\mathbb{Z},3)\to X\to K(\mathbb{Z}/2\mathbb{Z},...
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How did Thom calculate $MSO(k)$ using Silber's polyhedron in his 1954's paper?
I am now reading Thom's famous paper Quelques propriétés globales des variétés différentiables. In page 48, Thom used an auxiliary space $K$, which is a principal fiber bundle with base space $K(\...
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Cohomology operations are group homomorphisms
Let our spaces have the homotopy type of a CW complex, and let $E^*,F^*$ be two cohomology theories. A (degree $n$, stable) cohomology operation is a map $\Phi: E^q(X) \to F^{q+n}(X)$ for each $q$, ...
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Do Steenrod Squares have naturality with homomorphisms that don't come from continous maps
I was reading about the Steenrod Squaring operations in Milnor and Stasheff's, Characteristic Classes, now there is an axiom regarding naturality that says that given a continous map $f:(X,Y)\...
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What is the Steenrod algebra for finite fields?
I understand that the Steenrod algebra for finite fields with $p$ elements ($p$ prime) is understood, but do we know what the Steenrod algebra is for all finite fields?
Namely, what is $H\mathbb F_{p^...
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Stable cohomology operation and Steenrod algebra
In the Fomenko's Book,
we find the characterization of stable cohomology operations $O^S(q,G,H)$ as the projective limit of $$\cdots \longrightarrow \mathcal{H}^{q+n+1}(K(G,n+1);H)\longrightarrow\...
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Suspension map and Stable cohomology operation as inverse limit.
I found in the Fomenko's book that the stable cohomology operation is an inverse limit of $(H^{n+q}(K(G,n);H),f_n)$, where $G,H$ are a group (or rings, or fields for simplicity, it doesn't matter).
My ...
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On how to compute $H_*(\Omega^2 S^{n+2}, \mathbb Z/2)$ and its Steenrod operations
I've been trying to understand the calculation of $H_*(\Omega^2 S^{n+2},\mathbb Z/2)$ as an algebra, together with the dual Steenrod operations on it which lower the degrees. I've spent many hours ...
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Which integral Steifel-Whitney classes are universally $0$?
Let $BO(n)$ denote the classifying space of the orthogonal group $O(n)$. Then there is the well-known ring isomorphism
$$H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}/2[w_1,\dots,w_n] $$
where $w_i \in H^...
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Is it possible to find a torsion-free space with non-trivial $\operatorname{Sq}^{2^n}$
Is it possible to find a torsion-free space with non-trivial $\operatorname{Sq}^{2^n}$ ?
Torsion-free means integral cohomology is torsion-free. When $n \leq 3$, such spaces are projective space $\...
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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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Group homomorphism induces cohomology homomorphism
Let $\phi:G\to H$ a group homomorphism. I'm interested in knowing what can be said about $H^*(X;G)$ and $H^*(X;H)$ (singular cohomology) in terms of $\phi$.
One can define $\phi^*:C^n(X;G)\to C^n(X;...
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Reference request: The Steenrod squares generate all stable cohomology operations
The isomorphy of the Steenrod algebra to the cohomology of an Eilenberg-MacLane spectrum is a direct corollary of Brown representability and the fact, that all stable cohomology operations are ...
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How can I determine the Steenrod Square $Sq^2$ for complex projective space?
I am trying to learn about Steenrod Squares for algebraic varieties so that I can compute examples of complex topological K-theory using the Atiyah-Hirzebruch Spectral Sequence (AHSS).
One of the key ...
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Intuition behind cohomology operations
I try to unterstand cohomology operations, but I cannot get the intuition behind it. Could someone explain the intuition behind it?
My background: I have a basic understanding of homology and ...
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Applications of Steenrod Squares
I am reading through Glen Bredon's book "Geometry and Topology" and after giving the axioms for Steenrod squares plus the Adem relation, he proves that if $i$ is not a power of $2$ then $Sq^i$ is ...
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Closed manifolds with isomorphic cohomology rings, but different cohomology modules over the Steenrod algebra
For any $n > 2$, $\mathbb{CP}^n/\mathbb{CP}^{n-2}$ and $S^{2n}\vee S^{2n-2}$ have the same cohomology groups: for any ring $R$, we have
$$H^k(\mathbb{CP}^n/\mathbb{CP}^{n-2}; R) \cong H^k(S^{2n}\...
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Some clarifications about the Secondary Cohomology Operation associated to $Sq^2\circ Sq^2=0$
As explained in the title, I'm looking for some clarifications about the secondary cohomology operation associated to the relation $Sq^2\circ Sq^2=0$. I've just started reading the relevant chapter in ...
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Expositions of Postkinov Towers
I am giving a lecture on Postkinov towers and I want to teach the students a lesson :). These students have seen local coefficients, spectral sequences and cohomology operations at the level of ...
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Characterisation of Stable Cohomological Operations: $\Sigma (\tau_n (\imath_{A,n}))=\rho_{n+1}^*(\tau_{n+1}(\imath_{A,n+1}))$
I've started studying (stable) cohomological operations on my lecture notes, and I was given that an equivalent definition for a family of cohomological operations to be a stable cohomological ...
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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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Prove that Steenrod squares are stable
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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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 ...
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