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Tagged with steenrod-algebra at.algebraic-topology
12 questions with no upvoted or accepted answers
18
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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 ...
- 35.9k
13
votes
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
9
votes
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:
$$D_{2n} = \langle x,y ...
- 2,018
7
votes
0
answers
424
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kernel of the mod $2$ Bockstein on the first cohomology group
Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...
- 1,990
7
votes
0
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430
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algebraic structure of Integral Steenrod squares
It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations
$$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$
In the case where $a$ is odd, one can define an ...
- 928
6
votes
0
answers
210
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Generators for unitary bordism ring $\pi_*(MU)$
I’m reading Pengelley’s paper “The mod 2 homology of $MSO$ and $MSU$ as $\mathfrak A^*$ comodule algebras, and the cobordism ring”.
He has chosen very special generators $z_n \in H_n(MO; \mathbb F_2)$...
user131113
5
votes
0
answers
185
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Dual Steenrod squares
Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
- 1,623
5
votes
0
answers
225
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Generalizing the formula between Wu class and the Steenrod square
I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy
$$
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}$$
...
- 10.5k
4
votes
0
answers
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Is there a local simplicial formula for the Steenrod squares which commutes with the derivative on cochain level?
There is a well-known formula for the cup product of an $i$-cochain $A$ and $j$-cochain $B$ in simplicial homology given by
$$(A\cup B)(0\ldots i+j) = A(0\ldots i) B(0\ldots j)\;.$$
This formula ...
- 3,001
4
votes
0
answers
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
3
votes
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
3
votes
0
answers
180
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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 $...
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