Questions tagged [steenrod-algebra]

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1answer
113 views

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 ...
4
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1answer
213 views

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 ...
5
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0answers
168 views

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 $z_n \in H_n(MO; \mathbb F_2)$...
7
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2answers
305 views

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 ...
3
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0answers
250 views

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(...
5
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1answer
231 views

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(...
2
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0answers
82 views

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 ...
2
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0answers
74 views

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\...
2
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1answer
80 views

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{...
5
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1answer
205 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 ...
3
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0answers
114 views

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 ...
4
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1answer
136 views

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 ...
5
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0answers
180 views

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 $$ Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) . \tag{eq.1}$$ ...
11
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1answer
459 views

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 ...
6
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1answer
256 views

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 ...
5
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1answer
141 views

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 $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$ ...
3
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1answer
212 views

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}_p\to\...
20
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1answer
713 views

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....
11
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0answers
177 views

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//...
17
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1answer
580 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 ...
8
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1answer
263 views

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 ...
12
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2answers
401 views

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 ...
5
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2answers
482 views

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;\...
5
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1answer
456 views

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 ...
1
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1answer
99 views

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 ...
15
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1answer
423 views

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: ...
9
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0answers
178 views

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 ...
6
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2answers
391 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 ...
17
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2answers
624 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 ...
24
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2answers
2k views

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,...
7
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0answers
246 views

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 ...
12
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0answers
387 views

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 ...
4
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0answers
320 views

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 ...
4
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1answer
356 views

Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this. I wonder if there is any source telling about a ...
3
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1answer
153 views

cohomology ring of infinite iterated loop space

What is the cohomology ring $$ H^*(\Omega^\infty \Sigma^\infty (S^m\vee S^n);\mathbb{Z}_2)? $$ I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...
8
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2answers
780 views

cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...
9
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1answer
372 views

Steenrod operations on cohomology of grassmannians

Let $G_k(\mathbb{R}^n)$, $n\geq k$ and $G_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of ...
3
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0answers
174 views

Twisting of the power functor

Let $k$ be a field of characteristic $p$ and $D^b(k)$ be the infinity (equivalently, DG) category of perfect complexes over $k$. Let $C_p(=\mathbb{Z}/p)$ be the cyclic group on $p$ elements. For a $...
6
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3answers
745 views

Steenrod operations in algebraic geometry

What are some applications of Steenrod operations (or similar constructions) in algebraic geometry? I am dimly aware of the the use of these Voevodsky's work on motivic cohomology, and would be ...
6
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1answer
859 views

Do people still use Massey Products for computations in the Adams Spectral Sequence

Hey everyone, It seems to me like in the literature of the Adams Spectral Sequence, older publications (Toda, May, Tengora+Mahowald) make heavy and explicit use of Massey Products for computations. ...
16
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0answers
629 views

Steenrod algebra at a prime power

Let $n=p^k$ be a prime power. When $k=1$, the algebra of stable operations in mod $p$ cohomology is the Steenrod algebra $\mathcal{A}_p$. It has a nice description in terms of generators and ...
5
votes
2answers
1k views

Why are cup-i products and Steenrod Squares often (always?) unary?

One way to define the Steenrod Operations is to use the cup-i product, as in Mosher and Tangora's book. It basically says, given the chain complex from mod-2 homology $C_\ast$, define $D_0 : C_\ast\...
12
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4answers
968 views

$Sq^1$ cohomology of spaces

For any space $X$, the first Steenrod square cohomology operation $$Sq^1\colon H^\ast(X;\mathbb{Z}_2)\to H^{\ast +1}(X;\mathbb{Z}_2)$$ is a derivation, meaning that $Sq^1\circ Sq^1 = 0$ and $Sq^1(a\...
9
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3answers
2k views

Why does one consider the dual of the Steenrod algebra?

Why does one consider the dual of the Steenrod algebra?
4
votes
2answers
520 views

Adem-Wu relations from Bullett-Macdonald identities

Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...
67
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9answers
10k views

understanding Steenrod squares

There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...