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Questions tagged [homology-cohomology]

Use this tag if your question involves some type of (co)homology, including (but not limited to) simplicial, singular or group (co)homology. Consider the tag (homological-algebra) for more abstract aspects of (co)homology theory.

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Exercise in Simplicial Homology

In Basic Concepts of Algebraic Topology by Fred Croom, the homology groups of the $n$-skeleton of the closure of an $(n+1)$-simplex are computed in Theorem 2.9. (The geometric carrier of the complex ...
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Can you express this easy theorem in fancy categorical terms?

Here is a theorem (of homological algebra): Given $A \rightarrow B \rightarrow C$ in an abelian category $\mathcal{A}$. If for all $D \in \mathcal{A}$ we have that $Hom(D,A) \rightarrow Hom(...
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Homology groups of the set with Mayer-Veitoris

i need some help please my professor asked for homology groups of $S^2\cup d$ which $d={\{ (0,0,t) \mid -1 \leq t \leq 1}\} $ . We just find homology groups with some points and Mayor-...
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Explain projective representation vs faithful representation

This is very basic. I am trying to explain projective representation vs faithful representation in a most naive way to a class of middle school students. My formal way of understanding is that ...
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Certain Galois cohomology computation

Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \...
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Help with homology axioms

I am currently interested in the calculation $H_1(S^2-I)$ where $I$ is an interval embedded in the unit sphere. The answer should be 0, and Hatcher proves it in his book in the sections "Classical ...
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Semidirect product theorem to translate from german.

I need some help to translate this theorem and its proof, so I wanted to know if anyone could help me doing this. The book is Endliche Grouppen I, by Bertram Huppert, the theorem is the number 17.3, ...
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How to get the cohomology group in the exact sequence? $B_n^* \leftarrow Z_n^* \leftarrow H^n(C;G) \leftarrow B_{n-1}^* \leftarrow Z_{n-1}^*$

In Hatcher page 192, the sequence is extracted $B_n^* \leftarrow Z_n^* \leftarrow H^n(C;G) \leftarrow B_{n-1}^* \leftarrow Z_{n-1}^*$ from the diagram $Z_n^* = \ker \delta_n$, $B_n^* = Img \...
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Cellular action on CW complex

Let $G$ act cellularly on a CW complex $X$. For each $n\ge 0$, the action induces an action on the indexing set $I_n$ for the $n$-cells. Now look at the cellular chain complex $C_\bullet(X)$. Each ...
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A natural isomomorphism of $G$-modules

Let $A$ be a $G$ module, $\Bbb Z$ regarded as a trivial $\Bbb ZG$ module . Then $$ \Bbb Z \otimes _G A \rightarrow A/\mathcal{G}A, \quad m\otimes a \mapsto ma+ \mathcal{G}A $$ is an isomorphism. ...
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Existence of sphere maps of arbitrary degree

By definition, the degree of a map $f:S^n\rightarrow S^n$ where $n>0$ is defined to be $m$ in the equality $f(\omega)=m.\omega$, where $\omega$ is a generator of $H_n(S^n)$. I want to show the ...
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Borel subgroup of $SL_2(\mathbb{Z})$

As the title indicates, I want to ask what is the Borel subgroup of $SL_2(\mathbb{Z})$? I believe I read about it in one of James Milne's notes. But now I cannot find it. That is why I want to ask.
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Homology of the $n$-sphere

I know the computation of $H_*(S^n)$, but I do not understand what the following theorem means: “For any integer $n\ge 0$, $H_*(S^n)$ is a free abelian group with two generators, one in dimension $0$ ...
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Chain complex homotopy equivalent to its homology

Let $(C_*,d)$ be a chain complex of vector spaces over a field $F$. Can always be constructed a homotopy equivalence $C_* \rightarrow H(C_*)$ ? (Here $H(C_*)$ is seen as a chain complex with $d=0$) ...
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Complexifying de Rham Complex

In Madsen's Calculus to Cohomology, it is defined $H^*(M; \Bbb C):= H^*(M) \otimes_{\Bbb R} \Bbb C$ I am curious, suppose we start with the de Rham complex, $D$, tensor it by $-\otimes_{\Bbb R} \...
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Homology group of $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$

My friend asks me how to compute the homology group of $X=SL(2,\mathbb{R})/SL(2,\mathbb{Z})$. It is not hard to see that $H_0(X)=\mathbb{Z}$, and for $q\ge 4$ we have $H_q(X)=0$. But I don't know ...
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Misunderstanding in definition of homology of groups

I am following Brown's 'Cohomology of groups' and the homology is defined as follows: Let $\cdots \rightarrow F_{n}\rightarrow F_{n-1}\rightarrow \cdots \rightarrow F_{1}\rightarrow F_{0}\rightarrow \...
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Ring isomorphism between $H^*(G_n(\Bbb F^\infty); R)$ and Ring of characteristic classes

Definitions: Define the following map: let $k \in H^m(G_n(\Bbb F^\infty ), R)$, and $\xi= (E,B,p)$ then define $k(\xi) = g^*(k) \in H^m(B,R)$, where $g$ is the induced map from the bundle map $(F,g):\...
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The ampleness of canonical sheaves and the proof of “$X \simeq \mathrm{Proj}\left(\bigoplus_k H^0(X, \omega_X^k)\right)$”.

In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the ...
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Is there an algorithmic way to compute the quotient of two subgroups of $\mathbb Z^n$?

In algebraic topology one sometimes has to compute the homology groups via CW complexes, reducing the problem to the calculation of $Z/B$ where $B \subseteq Z \subseteq \mathbb{Z}^n $. It is difficult ...
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1answer
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Connected Hausdorff space all whose $n$-th homology groups, for $n\ge 1$, are trivial

Let $X$ be a connected Hausdorff space such that $H_n(X,\mathbb Z)=0, \forall n \ge 1$. Then does that necessarily imply that $X$ is path connected i.e. $H_0(X,\mathbb Z)=\mathbb Z$ ?
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On topological space whose homology groups/modules are trivial

Let $X$ be a simply connected topological space. Consider the following three statements : (1) $X$ is contractible. (2) For every commutative ring $R$, $H_0(X,R)=R$ and $H_n(X,R)=0, \forall n \ge 1$...
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Why homotopic equivalence is equivalent to quasi iso in K(I)

Let $X,Y \in K(A)$ where $A$ is an abelian category and $X,Y$ are complexes s.t. $X^i$ and $Y^i$ are injecive for every i. How can I prove that if $t : X \to Y$ is a quasi-isomorphism he $t$ is an ...
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Lusternik-Schnirelmann category of $\mathbb{RP}^2 = 3$ (or $2$ depending on the definition) using only homology

I want to prove that $LS(\mathbb{RP}^2)=3$, where $$LS(X) = \min{\lbrace k \in \mathbb{N} \hspace{3pt} | \hspace{3pt} \text{there exists an open cover }U_{1},...,U_{k} \text{ s.t. } i_{l}:U_{l} \...
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1answer
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Is repeating a loop twice the same as multiplication by $2$ in singular homology?

I have the following problem understanding the notion of coefficients in Singular Homology. Let $X$ be a topological space (for example closed differential manifold). Let $x:[0,1]\rightarrow X$ be a ...
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How do you call groups formed as direct sums of cyclic groups?

What do you call groups of the form $\mathbb{Z}\oplus\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/\mathbb{Z}d_1\oplus\mathbb{Z}/\mathbb{Z}d_2\oplus\cdots$ These kind of groups usualy appear as homology ...
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Wedge axiom of a homology theory like functor

Let $h_n\colon CW_*\to Ab (n\in \mathbb{Z})$ be a covariant homotopy invariant functor that sends cofibre sequences to exact sequences and is equipped with natural suspension isomorphisms. Then the ...
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Cohomology Operations

What are prerequisites for studying Cohomology Operations (or for studying Mosher & Tangora 's book on Cohomology Operations)?
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cohomological dimension of groups vs cohomological dimension of subgroups

Let $\Gamma$ be a group and $\Gamma^\prime$ a subgroup. Then, $\text{cd }\Gamma^\prime \leq \text{cd } \Gamma$ because a projective resolution of $\mathbb{Z}$ over $\mathbb{Z}\Gamma$ can also be ...
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For a group homomorphism $\alpha : G\rightarrow G'$, does the pullback $\alpha^\# : Mod(G')\rightarrow Mod(G)$ send $G'$-projectives to $G$-acyclics?

Let $\alpha : G\rightarrow G'$ be a group homomorphism. There is a natural functor $\alpha^\# : Mod(G')\rightarrow Mod(G)$ sending a $G'$-module $M$ to the $G$-module given by the same underlying ...
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Removing Homology Groups

I was trying to construct a space that has first $n$ homology groups any given abelian groups $G_1, ..., G_n$. To show this I would like to be able to do the following: Given any space $X$, I can form ...
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1answer
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Using simplicial homology to compute topological spaces homeomorphic to the quotient spaces of polygons

I am reading Introduction to Algebraic Topology by Rotman and the following is presented as a method to use simplicial homology to compute topological spaces that are homeomorphic to the quotient ...
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1answer
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Changing coefficients of cohomology and pullbacks

If I have a compact complex manifold $M$ and a map $f:M\to M$. Let $\mathbb F$ a field. Then $H^k(M;\mathbb F) $ is a $n$-dimensional vector space and chosing a basis, $f^*_{\mathbb F}$ defines a ...
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1answer
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Isomorphism between torsion subgroups of $H_q (M)$ and $H_{n-q-1} (M)$ where $M$ is a compact oriented $n$-manifold

It it the exercise in Massey's book Massey, William S., Singular homology theory, Graduate Texts in Mathematics, 70. New York Heidelberg Berlin: Springer- Verlag. XII, 265 p. DM 49.50; $ 29.20 (1980)...
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1answer
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Whether the induced map in de Rham cohomology is injective

Let $M, N$ be smooth manifolds, and let $f: M \rightarrow N$ be a surjective submersion, i.e. a surjective smooth map such that the differential $f_{*}$ is also surjective. I have shown that for all $...
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singular cohomology and Poincaré duality

Suppose $M$ is a n-dimensional, finite type, oriented, smooth manifold. A $k$-dimensional cycle in $M$ is a pair ($S$,$\phi$), where $S$ is a compact, oriented $k$-dimensional manifold without ...
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Computing the first simplical homology group of the torus $H_1(T)$

Let $K$ be the following triangulation of the torus. This triangulation of $T$ has $18$ $2$-simplexes; $27$ $1$-simplexes and $9$ vertices. Now using singular homology, the fact that singular ...
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1answer
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A false reasoning for a better intuition behind the universal coefficient theorem

Let $X$ be a topological space. I am trying to understand the natural homomorphism $$H^*(M,\mathbb{Z}) \to H^*(M,\mathbb{R}),$$ induced by the inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$. The ...
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1answer
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Rescaling a symplectic form and integral cohomology

Let $(M,\omega)$ be a symplectic manifold. I am trying to understand a procedure which seems so obvious that its implications are omitted in any article I could read. I encountered the following: ...
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chain homotopy equivalence and quasi-isomorphism

Suppose $(C,d)$ and $(D,\delta)$ are two chain complexes over a field and $f:C\to D$ is a chain map. We say $f$ is a quasi-isomorphism if it induces an isomorphism of the homology groups $H(C,d)\to ...
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Why is induced map on zero homology the identity and not negative the identity?

Suppose we have a simplicial map $f$ on a path connected simplicial complex $X$. The answer here: Induced map on zeroth homology is zero claims that the induced map on the $0$-homology given by $f_*: ...
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Existence of topological space which has no “square-root” but whose “cube” has a “square-root”

Does there exist a topological space $X$ such that $X \ncong Y\times Y$ for every topological space $Y$ but $$X\times X \times X \cong Z\times Z$$ for some topological space $Z$ ? Here $\cong$ means ...
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Group cohomology topologically with simplicial sets

I have a question about the usual formula for the differential in the usual projective resolution of $\mathbb{Z}$ as a $G$-module for a finite group $G$ Recall that for a $G$-module $A$, $C^i(G,A)$ ...
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2answers
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Existence of a type of simplicial complex

I want to prove that the following proposition is false: There exists a homologically trivial finite 2 dimensional simplicial complex $\mathcal K$ such that every edge (1 dimensional simplex) has at ...
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1answer
104 views

Homology of subset of orbit space

Assume a finite group $G$ acts on a topological space $X$ and $A\subseteq X$. Denote by $q$ the quotient map from $X$ to the orbit space $X/G$ (we take the quotient topology). Moreover, let $H_n(A)=0$ ...
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Reduced homology of a point is trivial from axioms

In the section Axioms for homology from Hatcher's Algebraic Topology (page 161) he says: Note that $\tilde{h}_n(x_0) = 0$ for all $n$, as can be seen by looking at the long exact sequence of ...
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Killing homology below middle dimension with equivariant surgery

Assume a finite group acts smoothly on a manifold $M$ of dimension $n$. Suppose $a\in H_i(M)$, where $i=1,\ldots,[n/2]$. Is there a way to kill $a$ with equivariant surgery and keep the same fixed ...
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1answer
44 views

Acyclic compact Lie groups of dimension 3

Are there any examples of compact connected Lie groups with vanishing first homology groups in dimension $3$ different from $S^3$?
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Computing the induced map on homology from projective space

Define a map from $F:P^1(\Bbb C)\times P^1(\Bbb C) \rightarrow P^2(\Bbb C)$ by $((x,y), (z,w)) \mapsto (xz, xw + yz, yw)$ What is the induced map on homologies? $$H_p(\Bbb{CP}^1 \times \Bbb{CP}^1) \...
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1answer
116 views

Show that the inclusion of the real projective plane in the complex projective plane is not null-homotopic?

In other words, how to show that $\mathbb{RP}^2$ is not contractible in $\mathbb{CP}^2$. Any hints would be appreciated.