Skip to main content

Questions tagged [cohomology-operations]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
10 votes
1 answer
321 views

Steenrod powers of the Thom class

René Thom in 1952 proved the formula $$ Sq^i(U_2)=\Phi_2(w_i), $$ which in modern parlance says that the Steenrod squares of the mod $2$ Thom class of an orthogonal bundle are the images under the mod ...
Mark Grant's user avatar
  • 35.4k
4 votes
0 answers
156 views

Is there any results about the stable (or unstable) cohomology operations on cohomology of Lie groups?

$\DeclareMathOperator\SU{SU}$For the $\mod p$ singular cohomology of classical Lie groups, such as $H^*(\SU(n); \mathbb{Z}/p\mathbb{Z})$, there are well known results about the actions of the stable ...
cyber's user avatar
  • 71
4 votes
1 answer
331 views

When the Pontryagin square is an even class?

Let $n$ be an even integer and $X$ a manifold. Given a cohomology class $B \in H^k(X,\mathbb{Z}_n)$, the Pontryagin square is a class $\mathfrak{P}(B)\in H^{2k}(X,\mathbb{Z}_{2n})$. Is it true that if ...
Andrea Antinucci's user avatar
7 votes
1 answer
237 views

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 ...
Shivang's user avatar
  • 71
8 votes
0 answers
138 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 ...
HenrikRüping's user avatar
6 votes
0 answers
211 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 ...
Mark Grant's user avatar
  • 35.4k
10 votes
1 answer
908 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 ...
Tim Campion's user avatar
  • 62k
23 votes
2 answers
762 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)...
Michael Albanese's user avatar
5 votes
1 answer
356 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 ...
FKranhold's user avatar
  • 1,623
2 votes
0 answers
104 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 ...
xir's user avatar
  • 1,984
18 votes
1 answer
916 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 ...
KotelKanim's user avatar
  • 2,280
17 votes
2 answers
742 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 ...
Bogdan's user avatar
  • 335
3 votes
0 answers
229 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 ...
user43326's user avatar
  • 3,041
5 votes
0 answers
318 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 ...
user43326's user avatar
  • 3,041
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 $\mathbb{Z}$ (...
emmagvr's user avatar
  • 133
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} \; ...
Samuel Monnier's user avatar
59 votes
3 answers
5k 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)...
Chris Gerig's user avatar
  • 17.2k
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 ...
Chris Gerig's user avatar
  • 17.2k
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 $(\...
Akhil Mathew's user avatar
  • 25.3k