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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An application of excision
Let $m\geq 1$ and $K\subseteq \mathring{\mathbb{D}^m}$ a convex, compact subset. Here $\mathring{\mathbb{D}^m}$ is the interior of the m-dimensional disk. Let $x\in K$.
I am trying to argue that for $...
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Are $S(S^n\times S^m)$ and $S(S^n \vee S^m \vee S^{n+m})$ homotopy equivalent?
For $n,m\geq 1$ the spaces $X=S^n\times S^m$ and $Y=S^n \vee S^m \vee S^{n+m}$ have isomorphic homology and cohomology groups. The cohomology groups are given by $H^i(X)=H^i(Y)=\mathbb{Z}$ for $i=0,n,...
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Graded-commutativity of cup product: Non-commuative coefficient ring
For $R$ a commutative ring and $X$ a topological space the cup product on the singular cohomology $H^{\ast}(X)$ is graded commutative. I have a question about the proof of this claim.
Some definitions ...
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Model of a chain map?
Let $R$ be a commutative unital ring with multiplication $\mu: R\otimes R \rightarrow R$.
Given topological spaces $X,Y$ and open subsets $A\subseteq X$, $B\subseteq Y$, pick a chain homotopy ...
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Cohomology determining homotopy type
Let $X$ be a path-connected topological space such that its reduced cohomology vanishes, i.e. $\tilde{H}^i(X)=0$. Does it follow that $X$ is contractible (i.e homotopy equivalent to a point)?
(This ...
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Defintion of relative cohomology cross product
This is a passage from my notes which lacks some details:
Let $X,Y$ be topological spaces, and $A\subseteq X$ and $B\subseteq Y$ be open subsets. Then there is a natural map $K\colon S_{\ast}(X,A)\...
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Vanishing of $r$-fold cup-products
In my notes there is the following result:
Suppose $X$ can be covered by $r$-many open and path-connected sets $X=X_1\cup\ldots \cup X_r$. Assume that $H^\ast(X_i)=0$ for all $i=1,\ldots, r$. Then ...
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Cup product structure on $\mathbb{R}P^2$
I don't understand a passage in my lecture notes. Here is the passage with my questions added in italics:
$H^\ast(\mathbb{R}P^2;\mathbb{Z}/2\mathbb{Z})$ is $\mathbb{Z}/2\mathbb{Z}$ in degrees $1$ and ...
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Cohomology ring of symmetric products (of manifolds)
Let $S_g$ be a closed, orientable surface of genus $g$ (new notation in light of the first comment). I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring ...
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How to compute the monomial matrices?
In GTM227 Combinatorial Commutative Algebra, Miller defined the monomial matrix to represent a map between two $\mathbb{N}^n$-graded free $S$ modules. Its columns are labeled by sourse degrees $a_p$ ...
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Examples of CW-complexes wich are 1-acyclic but no simply conected.
Hurewicz theorem states that if $X$ is a simply-connected CW-complex then $X$ is $(n-1)$-connected if and only if it is $(n-1)$-acyclic and that in this case $\pi_n(X)=H_n(X)$. Moreover, it is also ...
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The complementaries of homologous objects in $S^n$ are homologous?
Suppose $A$ and $B$ are two subsets of the $n$-sphere $S^n$ that have isomorphic finitely-generated homology groups. Do $S^n\setminus A$ and $S^n\setminus B$ also have isomorphic homology groups?
I ...
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Chain map sequence $0 \to H(C) \to C/B(C) \xrightarrow{d} Z(C)[-1] \to H(C)[-1] \to 0$ is exact for a cochain complex of $R$-modules?
Specifically, Weibel Page 10 Exercise 1.2.7b.
I think I proved Exercise 1.2.7a or that there exists an SES:
$0 \to Z(C) \to C \xrightarrow{d} B(C)[-1] \to 0$
given a complex of $R$-modules $C^{\cdot}$ ...
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Relationship between compactly supported homology and homology with orientation sheaf
Let $M$ be noncompact and non-orientable manifold of dimension d. Is it true that $H_{c}^{d-i}(M;Q)$ and $H^{i}(M;Q^{w})$ are isomorphic, where $Q^{w}$ is orientation sheaf and $Q$ is a field of ...
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Does path connectedness simplify the proof of the LES in reduced homology?
I'm reading Weintraub's Fundamentals of Algebraic Topology, in which there is an exercise (3.4.6 in the first edition) that wants us to show that for every path connected subspace $A$ of a path ...
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What is the motivation to introduce Tate-cohomology groups?
What is the motivation to introduce Tate-cohomology groups ?
Let $G$ be a Galois group and $M$ be a $G-$module.
Let $H^n(G,M)$ be usual Galois cohomology.
In group cohomology theory, we often ...
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What was the intended, more elementary solution to Hatcher $2B.5$? On isomorphisms in homology between a sphere complement and a subsphere
I solved Hatcher's exercise $2B.5$ but I wonder if there is a more elementary approach. Paraphrasing, and removing trivial or vacuous cases, the exercise is this:
Suppose $n\ge1$ and $0\le k\le n-1$ ...
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Third cohomology of a group
I'm familiar with two primary ways to discuss the n$^{th}$ cohomology of groups with coefficients in an abelian group $A$:
(1) Through the exploration of n-fold extensions.
(2) By examining the map $...
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Trouble understanding torsion
I'm reading this paper. In definition 6.1.2 it is mentioned that $H_n(G;A) \cong \text{Tor}_n^{\mathbb{Z}G}(\mathbb{Z}, A)$ (for all $n=0, 1, 2, ...$). What exactly is meant by the right-hand side? I'...
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$p$-torsion in fundamental group of Lie group.
I'm study theory of Lie groups and the question is: can Lie group G have $p$-torsion for prime $p \neq 2$ in its fundamental group $π_1(G)$?
I know standart examples of Lie groups like $\mathrm{O}_n(ℝ)...
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Stallings' Theorem
I am trying to understand Stallings' Theorem for lower central series. Here is the statement:
Say we have groups $A, B$ with lower central series $A=A_1, A_2, ...$ and $B=B_1, B_2, ...$ respectively. ...
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What is the most diagrammatic proof that $gf = 0 \implies \text{im } f \subset \ker g$?
The proof of the following fact is trivial to most Arrow theorists / Linear algebraists, but I'm developing software that needs to "understand" in a sense this basic fact, because it is ...
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Separating subspheres from each other; what are the weakest hypotheses required in Hatcher exercise $2B.4$?
The task:
Take integers $p,q\ge1$ and define $S^{p-1}\subset S^{p+q-1}\subset\Bbb R^{p+q}$ to be the subsphere consisting of points of $S^{p+q-1}$ whose last $q$ coordinates are zero, and define $S^{...
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Acyclicity of co-chain complexes in terms of quasi-isomorphisms.
Weibel gives equivalent conditions for acyclicity of chain complexes, one of which is that the chain complex map $0\to C_.$ is a quasi-isomorphism (i.e. $H_n(0)=0\to H_n(C_.)$ is an isomorphism). ...
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Can I use Auslander-Buchsbaum formula for polynomial rings? [closed]
Let $R=k[x,y]$ be a polynomial ring over a field $k$ and set $T=R/\langle xy,y^2\rangle $. Let $f:R\to T$ be the natural ring epimorphism. Is $\operatorname{pd}_RT=2$?
we know that $\...
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About assertion in Thom isomorphism theorem
On page 114 of Milnor's book
https://aareyanmanzoor.github.io/assets/books/characteristic-classes.pdf
We have the next theorem
He says that $H^i(E,E_0,\mathbb{Z}_2)=0$ for $i<n$ as a consequence ...
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Alexander-Whitney map gives a coalgebra?
Let $R$ be a unital ring with multiplication $\mu\colon R\otimes R \rightarrow R$. Consider the category $\mathcal{Ch}(R-\text{mod})$ of chain complexes of $R$-modules.
This category becomes a ...
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Does the inclusion map $\iota: \mathbb{Z}\rightarrow \mathbb{R}$ induce an inclusion of co/homology groups?
Let $M$ be a smooth $n$-manifold, and $\iota: \mathbb{Z}\hookrightarrow \mathbb{R}$ be the natural inclusion homomorphism of abelian groups. Then, it is easy to check that we obtain a well defined ...
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Image of a subspace under the induced homology map: does $f(X)\subset Y$ imply $f_*H_1(X)\subset H_1(Y)$?
Assume we have topological spaces $X$ and $Y$ and let $f\colon X\to Y$ be a continuous map.
Does it follow that the image of the induced homomorphism $f_*: H_1(X)$ is a submodule of $H_1(Y)$? In other ...
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Prove the mod 2 Steenrod algebra has counit
Edit: $\epsilon$ must send $Sq^i$ to $0$ since it is the case where $\mathcal{A}_2$ is a graded algebra. The Wikipedia page presents the answer itself.
I'm trying to show that the mod $2$ Steenrod ...
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Homology of space resulting from gluing boundaries of two solid tori via certain map
I'd really appreciate some hints or suggestions on how to approach the following problem:
Let $D^2$ be the unit disk in $\mathbb{C}$, and let $S^1 = \partial D^2$. Let $ X = D^2 \times S^1 \times \{0,...
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If $f: D^n \to D^n$ is a homeomorphism, then $f(S^{n-1}) = S^{n-1}$ [duplicate]
I've come across an exercise while self-studying Algebraic Topology:
Prove that if $f: D^n \to D^n$ is a homeomorphism, then $f(S^{n-1}) = S^{n-1}$
I've got some background on Homology theory (I've ...
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Abelian group as second cohomology group of a pair G,M
I'm currently studying group cohomology and in particular group extension; I'm trying to figure out a solution to the following problem: let A an abelian group, is possible to find a group G and a G-...
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Intersection of minimal topological spaces
I am working on some problems in toric arrangements theory, particularly in the proof that the integer cohomology of the complement manifold of these arrangements is torsion-free.
First, let us recall ...
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Constructing a map between suspensions that induces isomorphism on homology groups
This question is similar but different from this
Let the spaces be locally compact, Hausdorff and based.
Problem: Construct a map $\Sigma (X \times Y) \to \Sigma X \vee \Sigma Y \vee \Sigma (X \wedge ...
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Computing Floer homology of knots, going from graph to homology (following Manolescu's high-school level slides)
Recently I came across the following expository slides of Ciprian Manolescu: The unknotting problem: a journey from elementary to advanced mathematics, talk for the high school students at the ...
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Quotient of cohomology groups with different coefficents
Let $M$ be a smooth manifold, (if necessary I'm ok with assuming that $M$ is four dimensional, orientable, and closed). I wish to understand the quotient:
$$Q=H^1(M;\mathbb{R})/H^1(M;\mathbb{Z})$$
I ...
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How to compute cellular homology of $\mathbb R P^n$
A while ago, I'd asked a related question here, which went unanswered probably because it was too broad a question.
Suppose that we have chain complex for a CW comples $X$: $\cdots\to C_{n+1}\to C_n\...
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Analogue of short exact - long exact sequence theorem in homology for longer sequences
this is my first question here so I apologies in advance if I will miss details.
With a short exact sequence of modules we can produce a long one with their tor derived functors.
I was wondering if ...
4
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A map between suspensions which is a homology isomorphism
Problem: $Y \subset X$ be a subcomplex of a C. W. complex. Suppose that there is a retraction $r: X \to Y$.
Prove that there is a map from $\Sigma X \to \Sigma Y \vee \Sigma(X/Y)$ which is a homology ...
- 3,563
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If $M$ is compact connected non-orientable 3-manifold, then $H_1(M)$ is infinite.
Let $M$ be a compact connected non-orientable 3-manifold. The goal is to show that $H_1(M)$, the first integral homology group, is infinite.
There is a proof of this assuming $\partial M = \emptyset$. ...
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Find a space with same homology and fundamental group as a torus
I am looking for $X$ which has the same integral homology and fundamental group as the torus, $T$, which is not homotopic equivalent to $T$.
At first, I considered $S^1\vee S^1 \vee S^2$, but $\pi_1(S^...
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Homology of exterior of a solid knot in $S^3$
Let $K$ be a knotted solid torus in $S^3$, $T$ be its boundary torus, and $X$ be its exterior, i.e. the closure of $S^3\setminus N$. The goal is to compute $H_n(X)$, the integral homology of $X$.
...
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Relation of the mapping class group and the symplectic group.
Generally, I am currently investigating automorphisms of Riemann surfaces and their non-trivial action on the first homology.
I wonder whether one can make explicit statements on the relationship ...
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Group cohomology of $C_4$ with a certain non-trivial coefficient module
Let us consider the polynomial ring $M=\mathbb{Z}[x,y]$ as a $\mathbb{Z}[C_4]$-module via the $C_4$ action given by $x\mapsto y \mapsto -x.$ Define a grading on $M$ by setting $|x|=|y|=2$. We denote ...
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If $f: \mathbb CP^2 \rightarrow \mathbb CP^2$ is a homeomorphism, then $f(\mathbb CP^1)$ intersects $\mathbb CP^1$ [closed]
Consider the standard embedding $\mathbb CP^1 \subseteq \mathbb CP^2$.
Let $f : \mathbb CP^2 \rightarrow \mathbb CP^2$ be a homeomorphism. The goal is to show that $f(\mathbb CP^1)$ always intersects $...
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Cap Product on Oriented Surface
I'm reading Hatcher page 241 example 3.31 and have some questions about the calculation. Let $M$ be oriented surface of genus $g$ represented by the standard polygon. Let $\varphi_i$ be dual ...
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$\mathbb Z_3$-orientable manifold is $\mathbb Z$-orientable.
Let $M$ be a $\mathbb Z_3$-orientable $n$-manifold. The goal is to show that $M$ is $\mathbb Z$-orientable.
Assume that $M$ is closed and connected. Suppose, for a contradiction, that $M$ is not $\...
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Prove/disprove that $X$ is a manifold
Let $X$ be the 2-dimensional CW complex where 2-cells are glued as follows:
The goal is to prove/disprove that $X$ is a manifold. Here's my attempt to disprove this.
Suppose $X$ is a manifold. In ...
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0
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Application of Whitehead's theorem to show homotopy equivalence
Let $X$ be a CW complex satisfying $\pi_0(X) = \pi_1(X) = 0$, $H_2(X) \cong \mathbb Z^2$, and $H_j(X) = 0$ for $j \ne 2$.
I am trying to prove that $X$ is homotopic equivalent to $S^2 \vee S^2$.
By ...
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