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3 votes
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Is restriction to $p$ Sylow subgroup $\text{res} : H^1(H,M)\to H^1(H_p,M)$ injective?

This is a general phenomenon. It follows fron the existence, for any $G$-module $M$ (with $G$ a finite group) and any subgroup $H \leq G$, of a natural corestriction map $Cores: H^{\ast}(H,M) \...
Aphelli's user avatar
  • 34.7k
0 votes

degree of a map in terms of fundamental classes confusion

I can see there are many answer that directly solve the doubt. But I would like to describe it for the cohomology with real coefficients. There is an alternative definition of the degree of maps using ...
Trishan Mondal's user avatar
0 votes

Global dimension of a ring and Ext functor

It turns out that the answer is yes! Claim: Let $R$ be a commutative ring and let $n\in \mathbb{N}$. The following statements are equivalent: i) Every $R$-module has projective dimension at most $n$. ...
Conjecture's user avatar
  • 3,250
2 votes

Where exactly are we using the fact that $f$ is an odd function in the proof of the Borsuk-Ulam Theorem?

Hatcher uses the fact that an odd function $f:\mathbb{S}^n\to\mathbb{S}^n$ maps antipodal points to antipodal points, i.e. given a point $a\in\mathbb{S}^n$, $f$ sends $a$ and $-a$ to $f(a)$ and $-f(a)=...
secretGarden's user avatar
1 vote
Accepted

$\bmod \mathcal{C}$ Vietoris-Begle Theorem

Since the notes link to you are quite brief on the properties of Serre classes, I'll try and be somewhat detailed. Recall that the edge homomorphism in question has the form $$ \pi_n\colon H_n(E) =...
Ben Steffan's user avatar
  • 4,104
7 votes
Accepted

Is every (finite) simplicial complex the nerve of some covering?

This is true (and the finiteness assumption is unnecessary): you can take $X$ to be the geometric realization of $\Delta$. For each vertex $x\in X$, associate to it the union $U_x$ of the interiors ...
Eric Wofsey's user avatar
0 votes

Generalizing the relative homology group of the solid torus relative to the hollow torus

When this happens, unfortunately you have to dig into at least one of the induced maps or use fancy technology like a K√ľnneth theorem. Consider the map $i:X\times \partial D^2\hookrightarrow X\times D^...
Wyatt Kuehster's user avatar
0 votes
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The natural map from compact vertical cohomology to de Rham cohomology is not injective

The map $\iota: H^*_{cv}(E) \rightarrow H^*(E)$ induced by inclusion of forms is neither injective nor surjective in general. You can consider these two examples. Let $\pi: E \rightarrow S^1$ be the ...
lmiglio's user avatar
  • 66
3 votes
Accepted

Spectral sequence of $\text { the fibration } E X \xrightarrow{\Omega X} X$

Since $EX$ is contractible, $E^\infty_{p, q}$ must be 0 for all $(p, q) \neq 0$; this is what is meant by "the $E^\infty$-term is zero." In particular, this means that for each $(p, q) \neq (...
Ben Steffan's user avatar
  • 4,104
3 votes
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A quick question on equivalence between cellular and singular homology

The diagram commutes. Hatcher does not define induced homomorphisms $f_* : H_n(X) \to H_n(Y)$ for cellular maps $f : X \to Y$, but it clear from the naturality properties of singular homology that $f$...
Paul Frost's user avatar
  • 77.8k
2 votes
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cohomological spectral sequence of a CW-complex

Consider the diagram with exact rows and vertical maps induced from the "inclusion" $(X, X_{p - 1}) \hookrightarrow (X, X_p)$, $n$ arbitrary. All groups here are free, so applying $\...
Ben Steffan's user avatar
  • 4,104
5 votes
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Adem relations of Steenrod algebra

The Steenrod algebra is an algebra over $\mathbb{F}_2$, so in particular the coefficients are elements of $\mathbb{F}_2$. In other words, $$ 2 \mathrm{Sq}^4 = 0 $$ and $$ 3 \mathrm{Sq}^5 = \...
Ben Steffan's user avatar
  • 4,104
4 votes
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Computing action on cohomology group induced by conjugation

So I have an indirect argument. Consider the exact sequence $$0\to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0.$$ Viewing all of them as a trivial $S_3$ module, we can take cohomology to ...
Snacc's user avatar
  • 2,195
4 votes
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$H^2 (\mathbb{R}^3)$? (or in general $H^2 (\mathbb{R}^{2n+1})$)?

Santharoubane has proved in $1983$ in his article "Cohomology of Heisenberg Lie algebras" that $$ \dim H^i(\mathfrak{h}_n(\Bbb R))=\binom{2n}{i}-\binom{2n}{i-2} $$ for all $0\le i\le n$. For ...
Dietrich Burde's user avatar
0 votes
Accepted

Is $\delta \Delta^{-1} d$ the identity operator?

The Laplacian may be inverted under the assumption of no harmonic appearing in the Hodge decomposition, or working in equivalence classes modulo harmonic forms. On co-exact forms the Laplacian acts as ...
Theo Diamantakis's user avatar
3 votes
Accepted

An auto-homeomorphism of a grid of balls maps the center to itself.

Lets introduce the following: Definition: a point $x\in X$ is called local cut-point if for any open neighbourhood $U$ of $x$, there is a connected open neighbourhood $V$ of $x$ such that $V\subseteq ...
freakish's user avatar
  • 44.3k
2 votes

Cycles around compacta and the global Cauchy theorem

The argument is sloppy since it allows $\Gamma$ to intersect $K$. A clean way to argue is as follows. Define $\eta$ as in Rudin's paper (i.e. every point of $K$ is at distance $\ge 2\eta$ from the ...
Moishe Kohan's user avatar

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