# Questions tagged [steenrod-algebra]

The tag has no usage guidance.

59 questions
Filter by
Sorted by
Tagged with
202 views

### Triviality of Steenrod operation on $\Sigma^{2k}H(r,s)$

I want to prove that any Steenrod operation for $k>0$ $$\phi: H^{2^j}(\Sigma^{2k}H(r,s)) \rightarrow H^{2^l}(\Sigma^{2k}H(r,s))$$ is zero. Recall that $H(r,s)$ is the Milnor manifold which can be ...
359 views

### 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 ...
364 views

### 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 ...
102 views

### 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 ...
329 views

### 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 ...
284 views

### 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 ...
327 views

### 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 ...
214 views

### 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 ...
506 views

144 views

### 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 ...
383 views

### 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 ...
205 views

### 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 ...
1 vote
168 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 ...
417 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 ... 206 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 $z_n \in H_n(MO; \mathbb F_2)$... 412 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 ...
310 views

94 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 ... 111 views

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 $$[S^n\overset x\to H\mathbb F_2 \wedge X\overset{Sq^r\wedge id}\to \Sigma^r H\... 4 votes 1 answer 188 views ### specific modules over the Steenrod algebra with one generator I'd be happy to clarify the following. Consider the module which is a quotient of the Steenrod algebra mod 2 by the left ideal generated by \operatorname{Sq}^1, \operatorname{Sq}^2, \operatorname{... 5 votes 1 answer 308 views ### Ádem relations for the Steenrod and the Dyer–Lashof algebra In this paper by Nondas Kechagias, the Steenrod algebra and the Dyer–Lashof algebra are compared. The rough difference ist: The Steenrod algebra arises by dividing out the “cohomological” Ádem ... 4 votes 0 answers 140 views ### Dual Steenrod squares Fix the ground ring \mathbb{F}_2 and let X be a space with finite homology. Then we have an isomorphism \Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle which allows us to define the dual ... 4 votes 1 answer 160 views ### Steenrod algebra: Ádem relations from Milnor product formula The question is how to deduce the Ádem relations from the Milnor product formula. Straightforward approach leads to certain relation on binomial coefficients mod p. Could anyone tell me if there is a ... 5 votes 0 answers 205 views ### Generalizing the formula between Wu class and the Steenrod square I know that on the tangent bundle of M^d, the corresponding Wu class and the Steenrod square satisfy$$ 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}$$... 11 votes 1 answer 677 views ### Power operations from a Tate construction In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on \mathbb{E}_{\infty}-algebras, one finds a construction of the power ... 7 votes 1 answer 293 views ### Why is \pi_{-*}F(H\mathbb{F}_p, H\mathbb{F}_p) the mod p Steenrod algebra? Why is \pi_{-*}F(H\mathbb{F}_p, H\mathbb{F}_p) the mod p Steenrod algebra? (This is quite a common statement, seen, for instance, in EKMM.) To be more precise, stable mod p cohomology ... 5 votes 1 answer 208 views ### Pontryagin square, Postnikov square and their consistency formulas \mathcal{P}_2 is Pontryagin square$$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).$$question (i)... 3 votes 1 answer 593 views ### Bockstein homomorphism and Square Operations: Their consistency formulas Here are various ways to define "Bockstein homomorphism:" Let \beta_p:H^*(-,\mathbb{Z}_p) \to H^{*+1}(-,\mathbb{Z}_p) be the Bockstein homomorphism associated to the extension$$\mathbb{Z}...
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....