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.
23
questions
4
votes
1answer
53 views
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 $\mathbb{CP}^n$, $\mathbb{HP}^n$ and the Cayley plane. $\operatorname{Sq}^{2^n}$ takes the ...
4
votes
0answers
56 views
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 ...
1
vote
1answer
46 views
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;...
1
vote
1answer
62 views
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 ...
4
votes
2answers
238 views
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 ...
5
votes
2answers
329 views
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 ...
2
votes
1answer
171 views
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 ...
4
votes
1answer
199 views
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}\...
2
votes
1answer
75 views
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 ...
1
vote
0answers
62 views
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 ...
5
votes
0answers
56 views
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 ...
3
votes
2answers
380 views
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 ...
3
votes
1answer
147 views
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 $n>0$ ...
6
votes
2answers
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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}...
3
votes
1answer
108 views
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 ...
9
votes
1answer
189 views
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}(...
6
votes
1answer
216 views
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 ...
25
votes
1answer
1k views
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 ...
6
votes
1answer
113 views
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) ...
14
votes
2answers
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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 ...
0
votes
1answer
139 views
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 ...
9
votes
0answers
613 views
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 ...
9
votes
0answers
261 views
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 ...
