Questions tagged [cohomology-operations]

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7 votes
1 answer
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Relation between cohomology operations and the Adams spectral sequence

$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Cone}{Cone}$ I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...
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8 votes
0 answers
125 views

When is an ideal in the cohomology ring the kernel of a map induced by a map of spaces?

Let $X$ be a space and $I$ be an ideal in the cohomology ring $H^*(X)$. I am interested in the question whether there is a map of spaces $Y\rightarrow X$ such that $I$ is the kernel of the induced map ...
6 votes
0 answers
173 views

Borel equivariant cohomology operations

Fix a group $G$. For an abelian coefficient group $A$ let $H^*_G(-;A):=H^*(EG\times_G-;A)$ be the Borel cohomology with $A$ coefficents. This is a functor from $G$-spaces to graded abelian groups ...
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10 votes
1 answer
737 views

Is every additive cohomology operation stable?

To start, let's work with mod $p$ cohomology $H\mathbb F_p$ where $p$ is a prime. Consider the following three things: The bigraded abelian group of all unstable cohomology operations, comprising all ...
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23 votes
2 answers
739 views

Does every map $K(\mathbb{Z}, n) \to K(\mathbb{Z}/m, n + k)$ factor through $K(\mathbb{Z}/m, n)$?

Recall that a cohomology operation is a natural transformation $H^n(-; \pi) \to H^{n+k}(-; G)$ defined on CW complexes. Does every cohomology operation $H^n(-; \mathbb{Z}) \to H^{n+k}(-; \mathbb{Z}/m)...
5 votes
1 answer
301 views

Ádem relations for the Steenrod and the Dyer–Lashof algebra

In this paper by Nondas Kechagias, the Steenrod algebra and the Dyer–Lashof algebra are compared. The rough difference ist: The Steenrod algebra arises by dividing out the “cohomological” Ádem ...
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2 votes
0 answers
103 views

When are cohomology operations determined by their action on coefficients?

It is well-known that K-theory operations are determined by the action on coefficients, but I don't know the right way to prove this fact, nor a reference for the same. On the other hand, clearly this ...
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17 votes
1 answer
809 views

Steenrod squares as power operations vs. as cohomomology operations

There are already several excellent questions and answers on MO regarding Steenrod squares, understanding them in various ways and relating them to power operations and I think I get this. Still, I am ...
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17 votes
2 answers
705 views

Massey products in the Steenrod algebra

When building $kU/2$ via its Postnikov tower, there are some interesting Massey products that show up in the Steenrod algebra, and I'd like to understand them. I bet these appear somewhere in the ...
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3 votes
0 answers
213 views

Eilenberg-Moore spectral sequence for path-loop fibration over Q\Sigma X (reference request)

Related to the question here, here is another question. Consider the kernel of the map $H_*(QY;Z/p)\rightarrow H_{*+1}(Q\Sigma Y;Z/p)$. restricted to $PH_*(QY)$, and let's say $Y$ itself is a ...
  • 3,011
5 votes
0 answers
297 views

The behaviour of the suspension homomorphism on $H_*(QX;Z/p)$ for odd $p$ (Reference request)

The mod $p$ homology of $QX=\Omega ^{\infty}\Sigma ^{\infty}X$ for connected $X$ was computed by Dyer-Lashof Homology of Iterated Loop Spaces, Amer. J. of Math., vol.84, No.1 pp 35-88. 1962 It follows ...
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6 votes
2 answers
1k views

Definition of E-infinity operad

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ (...
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6 votes
2 answers
1k views

Pontryagin square of Stiefel-Whitney classes and Pontryagin classes

On a four-manifold, there is apparently a relation between the first Pontryagin class modulo 4 and the Pontryagin square of the second Stiefel-Whitney class: $\mathfrak{P}(w_2) = p_1 \; {\rm mod} \; ...
59 votes
3 answers
4k views

Operations via Morse Theory

I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)...
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7 votes
2 answers
1k views

The Norm Map in (group) cohomology via classifying spaces

The well-known transfer map in group (co)homology can be defined with only homological algebra, or with algebraic topology via classifying spaces (group cohomology of $G$ is isomorphic to ordinary ...
  • 16.2k
24 votes
4 answers
3k views

What do cohomology operations have to do with the non-existence of commutative cochains over $\mathbb{Z}$?

Let $X$ be a topological space. In elementary algebraic topology, the cup product $\phi \cup \psi$ of cochains $\phi \in H^p(X), \psi \in H^q(X)$ is defined on a chain $\sigma \in C_{p+q}(X)$ by $(\...
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