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Results tagged with cohomology
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user 43326
A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
1
vote
Accepted
Proving the induced map on the cohomology is an isomorphism
By taking a look at the cohomology of $RP^{2d-1}$ and its $(d-1)$-st skelton (which becomes $(2d-1)$-st skelton after suspending d times), which is just $RP^{d-1}$ we also see that $j$ induces an isomorphism …
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5
votes
The homology of the universal covering space, why so difficult to compute
Denote $G=\pi _1(X)$. Then we have a fibration $\tilde{X}\to X \to BG$, which leads to the Eilenberg-Moore spectral sequence
$$ Tor ^{H^*(BG)}(H^*(X),H^*(pt))\Rightarrow H^*(\tilde{X})$$
provided tha …
- 3,051
5
votes
0
answers
214
views
The action of Steenrod squares on the indeterminate in Bullett-Macdonald identities
The Bullet-Macdonald identity (c.f. On the Adem relations)is the following:
$$P(s^2+st)P(t^2)=P(t^2+st)P(s^2).$$
where we have $P(s)=\Sigma _s s^iSq^i$. This is known to be equivalent to the
Adem rela …
- 3,051
4
votes
Accepted
group completion theorem of homology as Hopf algebras
List item $H_*(M)[\pi _0(M)^{-1}]$ inherits its coalgebra structure from
$H_*(M)$. i.e., there is a unique coalgebra structure so that the localization
becomes a map of coalgebras. Concretely we "e …
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1
vote
cohomology ring of the fundamental group of unordered configuration space
Question 1 Yes, at least up to extension problem. The proof of the fact that $Conf(S-Q_k,k)$ is $K(\pi ,1)$
provides an explicit decomposition of $Conf(S-Q_k,k)$ into a fibration of $K(\pi ',1)$'s. …
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1
vote
Accepted
mod p cohomology ring of alternating groups
Now the double coset formula for the compositions $BP\stackrel{Bi}{\rightarrow} B\Sigma _p \stackrel{tr}{\rightarrow}BP$ says that in mod $p$ cohomology, this composition is the sum
of maps induced by …
- 3,051
4
votes
cohomology of classifying space of permutation groups
We can, in theory, determine the image using the fact that
the Quillen's map is injective (as a matter of fact, isomorphic) for the symmetric groups, that is elements of mod $2$ cohomology of the symmetric …
- 3,051
2
votes
0
answers
191
views
The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request
All cohomology and homology will be $Z/2$ coefficient. … Theorem II.11, and in cohomology one gets
$$f(x_1,x_2)\rightarrow f(x_1,x_2)+f_(x_1+x_2,x_2)+f(x_1+x_2,x_1)$$
By ``dualizing'' in homology one gets
$$A(s)\cdot A(t)\rightarrow A(s)\cdot A(t) + A(s+t) …
- 3,051
5
votes
Decompose $MT(E(d)\times_{\mathbb Z_2} SU(2))$ as the wedge sum or smash product of spectra
Let $Y$ be a space, $V$ be a virtual bundle of dimension $0$ over $Y$ (this $V$ is your $-V$). Then $Thom(Y,2V)$ is almost never (except when $Y$ is contractible, or something like that) a smash prod …
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7
votes
Accepted
cohomology of iterated loop space on spheres
As is known from J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" Ann. of Math. (2), 81 : 2 (1965) pp. 211–264, the algebra structure of an underlying algebra of a Hopf algebra is quite li …
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