Questions tagged [steenrod-algebra]
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63 questions
2
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Higher Bockstein maps in group cohomology
Let $p$ be an odd prime and $n>1$. I am trying to understand why the cohomology ring $H^{\ast}(\mathbb{Z}/p^n;\mathbb{F_p})$ is given by
$$\mathbb{F}_p[y]\otimes\Lambda(x),$$
with $|x|=1,|y|=2$ and ...
- 245
6
votes
1
answer
348
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Detailed exposition of construction of Steenrod squares from Haynes Miller's book
$\DeclareMathOperator\Sq{Sq}$Chapter 75 of Haynes Miller's book on algebraic topology contains a beautiful construction of the Steenrod squares $\Sq^i$.
Roughly speaking, it goes as follows. All ...
- 63
8
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2
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600
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Derivations in the Steenrod algebra
Let $\mathcal A^\ast$ be the (mod 2) Steenrod algebra.
Question 1:
Is there a classification of homogeneous elements $D \in \mathcal A^n$ such that $D^2 = 0$?
Question 2: Is there a classification of ...
- 64k
7
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1
answer
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Generalisation of Hirsch formula for the associativity of Steenrod's higher $\cup_2$ product with $\cup_1$ and cup products
For $f$, $g$ and $h$ cochains, the Hirsch formula is given as
$$ (f\cup g)\cup_1 h=f\cup (g\cup_1 h)+(-1)^{q(r-1)}(f\cup_1 h)\cup g.$$
Is there a more general formula that relates the associativity of ...
- 71
3
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1
answer
260
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Cohomology ring $H^*(\operatorname{SL}(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$
$\DeclareMathOperator\SL{SL}$In Soulé's paper "The cohomology of $\SL_3(\mathbb{Z})$" the cohomology ring $H^*(\SL(3,\mathbb{Z}),\mathbb{Z})_{(2)}$ is determined in Theorem 4.iv. I'm wanting ...
- 545
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2
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409
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Triviality of Steenrod operation on $\Sigma^{2k}\mathbb{CP}^n$
I was going through this paper by Tanaka. I am actually stuck at Lemma 5.2, page 365, given below also
The argument he gives above works, in particular for $\operatorname{Sq}^{2^r-2^j}$ but I am not ...
10
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1
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418
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Are all degree-1 cohomology operations Bocksteins?
I'm interested in cohomology operations (in ordinary cohomology)
$$H^i(-, G)\rightarrow H^{i+1}(-, H)\;,$$
that is, elements of
$$H^{i+1}(K(G, i), H)\;.$$
I know that $K(G, 1)=BG$, so for $i=1$, those ...
- 3,001
4
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0
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120
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Is there a local simplicial formula for the Steenrod squares which commutes with the derivative on cochain level?
There is a well-known formula for the cup product of an $i$-cochain $A$ and $j$-cochain $B$ in simplicial homology given by
$$(A\cup B)(0\ldots i+j) = A(0\ldots i) B(0\ldots j)\;.$$
This formula ...
- 3,001
8
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1
answer
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Analogue of Bockstein for crossed module extensions and higher Steenrod square
It is well known that in $\mathbb{Z}_2$-valued simplicial cohomology (and other cohomologies)
$$ Sq^1 = \beta\;,$$
where $Sq^1$ is the first Steenrod square and $\beta$ is the Bockstein homomorphism ...
- 3,001
10
votes
2
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350
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What is an unstable dual-Steenrod comodule?
$\newcommand\Sq{\mathit{Sq}}$Recall that a (graded) module $V^\ast$ over the Steenrod algebra $\mathcal A^\ast$ is said to be unstable if $\Sq^i v = 0$ for $i > |v|$. The motivating example, of ...
- 64k
9
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2
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417
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Over which (graded) rings are all modules decomposable into indecomposables?
A module is decomposable if it is the direct sum of two modules. The process of splitting summands off of a decomposable module does not need to terminate, so infinitely generated modules do not ...
- 6,162
4
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0
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356
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Eilenberg-Maclane spectrum and $E_{\infty}$-algebra structure on singular cochain complex
I'm trying to understand how the $E_{\infty}$-algebra structure on the singular cochain complex $C^{\bullet}(X)$ of a topological space $X$, in at least somewhat down-to-earth terms. (I'm coming at ...
- 241
12
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2
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558
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Modules over the integral dual Steenrod algebra as linear functors
Let $\text{Latt}$ denote the category of lattices, i.e., finitely generated free abelian groups. In the appendix to Lecture 4 of Condensed.pdf, Scholze considers functors $F \colon \text{Latt} \to \...
- 5,504
8
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1
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Why does $\iota_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ not come from $H^8(K(\mathbb Z/2,4);\mathbb Z)$?
In Hatcher's Chapter 5 (https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf) on page 574 (page 57 in the pdf), he states that $\iota_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ is not in the image of $H^...
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1
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Wall's presentation of the Steenrod algebra
In the paper "Generators and Relations for the Steenrod Algebra" (C. T. C. Wall, Annals of Mathematics, Second Series, Vol. 72, No. 3 (Nov., 1960), pp. 429-444) Wall shows that there is a ...
- 1,554
5
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1
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Examples of non-zero negative Steenrod operations
In JP May's paper A general algebraic approach to Steenrod operations, Steenrod operations are constructed in wide generality. In this context, it is not necessarily true that negative Steenrod ...
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223
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Two $\mathbb Z$-algebra structures on $\mathbb Z\otimes_{\mathbb S} R$
$\newcommand{\Sph}{\mathbb S} \newcommand{\Z}{\mathbb Z} \newcommand{\F}{\mathbb F}$
In this question I will abuse notation by writing $A$ for the (generalized) Eilenberg-MacLane spectrum associated ...
- 15.9k
1
vote
1
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230
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Steenrod operations from the delooping viewpoint
Let $F$ be a finite field, $Sq^i$ be the $i$-th Steenrod operation
$$ H^*(-;F) \to H^{*+i}(-;F).$$
By Yoneda lemma, such operation is a map $\phi_i: B^{*}F \to B^{*+i} F$, where $B$ denotes the ...
- 5,230
7
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1
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477
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Action of Steenrod algebra on Chern classes
This is question about result from Brown and Peterson $H^*(MO)$ as an algebra over the Steenrod algebra. Unfortunately, the paper is not available on the Internet, so I can't find the proof.
One of ...
user131113
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0
answers
210
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Generators for unitary bordism ring $\pi_*(MU)$
I’m reading Pengelley’s paper “The mod 2 homology of $MSO$ and $MSU$ as $\mathfrak A^*$ comodule algebras, and the cobordism ring”.
He has chosen very special generators $z_n \in H_n(MO; \mathbb F_2)$...
user131113
7
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2
answers
494
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How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?
Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has ...
- 54.6k
3
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0
answers
345
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mod $p$ homology of Thom spectra MSU
Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big(...
user131113
5
votes
1
answer
386
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Why does the Steenrod algebra act faithfully on $H^\ast(BC_p)$?
Define the Steenrod algebra $A^\ast$ to be the algebra of all stable mod $p$ cohomology operations. Without actually computing $A^\ast$, is it possible to see that $A^\ast$ acts faithfully on $H^\ast(...
- 64k
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0
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108
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Computation of mod p homology of $MSU$
I am trying to proof Novikov theorem
\begin{equation}
MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i.
\end{equation}
This can be proved by using ...
user131113
2
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0
answers
127
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Stable homology operations
Let $x\in (H\mathbb F_2)_n(X)=[S^n,H\mathbb F_2 \wedge X]$ be a homology class for a space $X$. Is there a description of
$$[S^n\overset x\to H\mathbb F_2 \wedge X\overset{Sq^r\wedge id}\to \Sigma^r H\...
- 21
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1
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specific modules over the Steenrod algebra with one generator
I'd be happy to clarify the following. Consider the module which is a quotient of the Steenrod algebra mod $2$ by the left ideal generated by $\operatorname{Sq}^1, \operatorname{Sq}^2, \operatorname{...
- 171
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1
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386
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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 ...
- 1,623
5
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0
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185
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Dual Steenrod squares
Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
- 1,623
4
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1
answer
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Steenrod algebra: Ádem relations from Milnor product formula
The question is how to deduce the Ádem relations from the Milnor product formula. Straightforward approach leads to certain relation on binomial coefficients mod p.
Could anyone tell me if there is a ...
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5
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0
answers
225
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Generalizing the formula between Wu class and the Steenrod square
I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy
$$
Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) .
\tag{eq.1}$$
...
- 10.5k
12
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1
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Power operations from a Tate construction
In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $\mathbb{E}_{\infty}$-algebras, one finds a construction of the power ...
- 858
7
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1
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Why is $\pi_{-*}F(H\mathbb{F}_p, H\mathbb{F}_p)$ the mod $p$ Steenrod algebra?
Why is $\pi_{-*}F(H\mathbb{F}_p, H\mathbb{F}_p)$ the mod $p$ Steenrod algebra? (This is quite a common statement, seen, for instance, in EKMM.)
To be more precise, stable mod $p$ cohomology ...
- 765
5
votes
1
answer
263
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Pontryagin square, Postnikov square and their consistency formulas
$\mathcal{P}_2$ is Pontryagin square
$$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$
$\mathfrak{P}$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$
question (i)...
- 10.5k
3
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1
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691
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Bockstein homomorphism and Square Operations: Their consistency formulas
Here are various ways to define "Bockstein homomorphism:"
Let $\beta_p:H^*(-,\mathbb{Z}_p)
\to H^{*+1}(-,\mathbb{Z}_p)$ be the Bockstein homomorphism associated to the extension $$\mathbb{Z}...
- 10.5k
20
votes
1
answer
874
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Odd primary dual Steenrod algebra
My question is related to
this, this, and this older questions.
Let $\mathcal A_*$ be the dual Steenrod algebra.
This is a super-commutative Hopf algebra, and so its $Spec$ is an algebraic super-group....
- 43.2k
11
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0
answers
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What are examples of spectra whose mod 2 cohomology contain A//A(n)?
Let $//$ denote the Hopf algebra quotient. We know that:
$$HF_{2}^*(ko) \simeq A//A(1)$$
$$HF_2^*(tmf) \simeq A//A(2)$$
By Hopf invariant one, we know there is no $X$ such that $HF_2^*(X) \simeq A//...
- 3,539
19
votes
1
answer
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Steenrod squares as power operations vs. as cohomology 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 ...
- 2,320
8
votes
1
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363
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Adams spectral sequence and short exact sequences. Some clarifications
as the title suggests I'm looking for some clarifications in the computations of the ext charts of some $A(1)$-modules arising as extensions of other modules. In particular, I've the following example ...
- 503
12
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2
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597
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Steenrod powers of Pontryagin classes
It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true ...
- 1,452
8
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2
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723
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Adem relations of Steenrod square without modding out the coboundaries
In the paper Products of Cocycles and Extensions of Mappings,
Steenrod introduced the cup-$i$ product and Steenrod square $Sq^k$:
$$
Sq^k(x_n) \equiv x_n \smile_{n-k} x_n,\ \ \ x_n \in C^n(M^d;\...
- 4,826
7
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1
answer
964
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Associativity of Steenrod's cup-i product
In the paper Products of Cocycles and Extensions of Mappings,
Steenrod introduced the cup-i product (and Steenrod square). I would like to ask if Steenrod's cup-i product associative or not? The paper ...
- 4,826
3
votes
1
answer
215
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Cartan Formula for Steenrod square on cocycles
Let $x_n,y_n,\cdots$ be cocycles in $Z^n(X,\mathbb{Z}_2)$ (not cohomology classes in $H^n(X,\mathbb{Z}_2)$).
Let $Sq^k(x)\equiv x_n \cup_{n-k} x_n$ be the Steenrod square (This definition is valid for ...
- 4,826
18
votes
2
answers
617
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A cochain-level Adem relation?
The original paper on Steenrod squares, Steenrod's "Products of cocycles and extensions of mappings", 1947, uses an explicit combinatorial formula for the squares in terms of simplicial cochains: ...
- 54.6k
9
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0
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239
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The Steenrod Algebra of the Dihedral Group $D_{2n}$, $n=0 \pmod{4}$
As the tile suggests, I'm interested in computing the action of the Steenrod Algebra on $H^*(D_{2n};\mathbb{Z}_2)$, for $n=0 \pmod{4}$. Let us start with some definitions/facts:
$$D_{2n} = \langle x,y ...
- 2,018
6
votes
2
answers
570
views
An integral cohomology operation related to Steenrod square?
Let $\beta: H^n(X, \mathbb{Z}_2)\to H^{n+1}(X, \mathbb{Z})$ be the Bockstein homomorphism. Is it possible to define a cohomology operation $f: H^{n+1}(X, \mathbb{Z})\to H^{n+k+1}(X, \mathbb{Z})$ such ...
- 383
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2
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771
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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 ...
- 335
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votes
2
answers
2k
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Steenrod operations in etale cohomology?
For $X$ a topological space, from the short exact sequence
$$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0 $$
we get a Bockstein homomorphism
$$H^i(X,...
- 2,809
7
votes
0
answers
424
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kernel of the mod $2$ Bockstein on the first cohomology group
Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...
- 1,990
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0
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561
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When does an $E_\infty$ algebra come from a commutative differential graded algebra?
Suppose that $K$ is an $E_\infty$-algebra on a space $X$ (more generally, any ringed topos; also, feel free to assume that $X$ is a point). That is, $K$ is a cochain complex of sheaves on $X$, endowed ...
- 16.2k
7
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0
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430
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algebraic structure of Integral Steenrod squares
It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations
$$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$
In the case where $a$ is odd, one can define an ...
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