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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6 views

What conditions are sufficient for the Leray-Hirsch theorem to be a Künneth formula?

Given a fiber bundle $F \to E \to B$ over a paracompact base $B$, assume its cohomology satisfies all the required properties for the Leray-Hirsch theorem to hold. This tells us that $$ H^n (E,R) \...
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Using smith normal form in calculating the simplicial homology groups of the triangular parachute

I am trying to solve this question: Compute the simplicial homology groups of the triangular parachute obtained from $\Delta^2$ by identifying its three vertices to a single point. And I found this ...
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34 views

In which sense is cohomology dual to homology?

Colimits in a category $C$ are called "colimits" because they are limits in the dual category $C^\mathrm{op}$. Question: Why is cohomology called "cohomology"? In which way does it ...
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1answer
38 views

What is the bilinear intersection form of real homology of 4-manifold?

I am currently reading Invariants of 3-manifolds via link polynomials and quantum groups by N. Reshetikhin and V.G. Turaev. In section 3.2, given a framed link $L$, we can get a $4$-manifold $D_L$ by ...
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Confused about notation for the cohomology of Grassmannians

In Milnor and Stasheff's book "Characteristic Classes", problem 7b, they ask us to show that the cohomology ring $H^* (G_n (\mathbb{R}^{n+k}, \mathbb{Z}_2)$ is generated by the Stiefel-...
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1answer
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Is there an analogue of the loop space for homology?

Is there an endofunctor $U: \mathrm{Top} \to \mathrm{Top}$ (or from some good subcategory) such that $H_n(UX) = H_{n+1}(X)$ for any $n \geq 1$
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1answer
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Cup product pairing is nonsingular for closed R orientable manifolds when R = Z and torsion in $H^∗(M;Z)$ is factored out?

I was reading Hatcher on the cup product being nondegenerate. I could follow the proof in the case when $R$ is a field, but I am not sure how to interpret For field coefficients or for integer ...
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1answer
30 views

Show that $H_k(S^n,x) \to H_k(S^n,E^+_n)$ is isomorphic for all k [duplicate]

I am working on the proof that for all $k$ and for all $x \in S^n$ $$H_k(S^n,x) \to H_k(S^n,E^+_n)$$ is isomorphic, where we denote by $E^+_n$ the upper hemisphere of $S^n$. The proof use the long ...
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1answer
47 views

Cohomology of a blow up and Mayer-Vietoris

I have been reading the section on the cohomology of a blow-up in Griffths-Harris "Principles of algebraic geometry", and I am a bit confused about a result that their recipe, which uses the ...
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1answer
50 views

Is there a manifold $X$ with $H_3(X)\neq 0$ that can be embedded in $\mathbb{R}^3$?

I think there is no manifold $X$ with $H_1(X)\neq 0$ that can be embedded in $\mathbb{R}$, although I am not sure. I thought there is a manifold $X$ with $H_2(X)\neq 0$ that can be embedded in $\...
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Questions in the proof of $H_n(X)\simeq H_n(A)\oplus H_n(X, A)$

I am working on the following proof and have two questions on it: Theorem If $A\subset X$ is a retract of $X$ then, $$H_n(X)\simeq H_n(A)\oplus H_n(X, A),$$ all $n\geq 0$. Proof. Let $r:X\...
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Relation between reduced and unreduced cohomology

I have found a lemma in nlab I can't quite make sense of (http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/generalized+(Eilenberg-Steenrod)+cohomology#FromUnreducedToReducedCohomology). It ...
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Homology of $\mathbb{R}P^n \times \mathbb{R}P^m$

If we know the homology of $\mathbb{R}P^n$, how can we determine the homology of $\mathbb{R}P^n \times \mathbb{R}P^m$ (where $n$ and $m$ are natural numbers)?
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Action of ambient isotopies on the homology of an embedded $n$-torus in $\mathbb{R}^{n+1}$ and $S^{n+1}$ that bring the $n$-torus back to itself

Let $T^n$ be the $n$-torus embedded in $\mathbb{R}^{n+1}$. Consider the ambient isotopies that bring $T^n$ back to itself. These isotopies have an action on $H_1(T^n)$ described by some subgroup $G_n\...
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1answer
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$\mathscr{I}$ is injective iff for every $U \subseteq X$, $R \subseteq \mathbb{Z}_U$, and map $f: R \to \mathscr{I}$ extends to $\mathbb{Z}_U$.

This is Hartshorne III.2.6. I'm very stuck on how to proceed since the solutions online I usually refer to have an error here. Here $X$ is a noetherian topological space and $j: U \hookrightarrow X$ ...
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Dold-Thom stabilization and the inverse map on the homology.

Consider the closed immersion of complex varieties $i:\text{Sym}^n(\mathbb{P}^d_{\mathbb{C}})\hookrightarrow \text{Sym}^{n+1}(\mathbb{P}^d_{\mathbb{C}})$ induced by adding a fixed point. Let $Z$ be ...
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Why is this true: $\ker(d_{C,i-1})=H_i(C_*)$?

I try to understand the proof of the long exact sequence of homology. The proof uses the snake lemma, which is logic but I don't get one point: We end with the following diagram: Why do we have that $...
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Support of homology in quasi-projective varieties.

Given a quasi-projective complex variety $X$ and a positive integer $i<\text{dim}(X)-1$. Consider the homology group $H_i(X(\mathbb{C}))$. Is it possible to find a subvariety of codimension at ...
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Why is this diagram commutative?

I try to understand of the proof for the long exact sequence in homology (see here). Now I got stucked why the following diagram should be commutative. What do I oversee? Many thanks for some help!
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1answer
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The boundary of a $singular\; 0-simplex$

In Singular homology, Given a $singular\; n-simplex$, $\varphi$, we define the $singular\; (n-1)-simplex$, $\partial_i{\varphi}$, $$\partial_i{\varphi}(x_0,x_1,\dots,x_{n-1})=\varphi(x_0,x_1,\dots,x_{...
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Signed Morse homology of $\mathbb{R}P^2$

Here's my understanding of computing the signs in Morse homology (following the book by Audin and Damian). Let $f$ be a Morse function on a manifold $M$ with a negative pseudo-gradient $X$. Let $W^s(p)...
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1answer
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Reduce homology of $H_1(\mathbb{R}^n,x)$

Let $x\in\mathbb{R}^n$ and $n>1$. My goal is to compute $H_1(\mathbb{R}^n,x)$. I write the exact sequence $$H_1(\mathbb{R}^n)\to H_1(\mathbb{R}^n,x)\to \widetilde{H}_0(x)\to \widetilde{H}_0(\mathbb{...
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Compatibility and consistency between two definitions of cohomology in two books (about coboundary operators and 1-cocycles and computing cohomology)

I was reading cohomology from Neukirch's book, and there he referenced to Hall's book. The two approaches are almost the same (are they not?), and they should give us the same results (cohomology ...
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111 views

Functoriality of Gysin homomorphism defined via Thom construction

I want to understand how Gysin homomorphism (in multiplicative cohomology theories) for proper oriented maps between smooth manifolds works and in particular prove some facts about it. I am almost ...
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1answer
49 views

Two fold map while calculating Mayer–Vietoris sequence of $\mathbb RP^2$.

My reference answer is: https://math.stackexchange.com/a/768843/342943 My problem is: While writing MW exact sequence $$\cdots\to H_2(M)\oplus H_2(D^2)\to H_2(\mathbb{R}P^2)\to \underbrace{H_1(S^1)}_{\...
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1answer
56 views

Do open embeddings preserve the intersection pairing?

In the following, all (co-)homology is taken with $\mathbb{Z}$-coefficients. Let $M$ be a smooth, oriented, $n$-dimensional manifold. Then, there is a Poincaré duality isomorphism $D\colon H_c^{n-k}(M)...
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Hatcher exercise 43c, section 2.2: If $X$ is a CW complex with finitely many cells in each dimension, then $H_n(X;G)$ is a direct sum...

Below is the question. I will briefly describe how I solved parts a and b, then say why I'm stuck on c. There are other questions on this site which address this exercise, but none dealing with part c ...
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3answers
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Proof of $\tilde{H}_*(X) \cong H_*(X, x_0) $.

Let $X$ be a topological space and $x_0 \in X$. I want to show that the relative homology group, $H_*(X, x_0)$, is isomorphic to the reduced homology group, $\tilde{H}_*(X)$. By considering the long ...
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73 views

Explicit description of the homology of the n-sphere

I'm currently studying the homology of manifolds (so it's mainly orientation and duality questions) and I was wondering how to explicitly compute homology of spaces and particularly $n$-spheres. ...
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1answer
61 views

Conditions for null-homotopic chain map in a specific example

I am trying to calculate the ring $Ext^{\bullet}_R(k,k)$ where $R=k[x,y]/(xy)$ and $k$ is regarded as an $R$-module via $x$ and $y$ acting as zero. I thought I was done but then to my demise I found a ...
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1answer
35 views

Computation of $\tilde{H}_k(\bigvee_{j = 1}^N S^n)$.

In my algebraic topology course, we state the following proposition $$ \tilde{H}_k(\bigvee_{j = 1}^N S^n) \cong \begin{cases} \mathbb Z^N & \text{if } k = n,\\ 0 & \text{else}, \end{cases} $$ ...
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Proof of $\tilde{H}_k(X \lor Y) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y)$.

In my algebraic topology course, we showed the following statement: Let $X, Y$ two topological spaces, $(x_0, y_0) \in X \times Y$ and $X\lor Y$ the wedge product of $(X, x_0)$ and $(Y, y_0)$. If $x_0$...
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70 views

To understand whether a module is flat, injective or projective through a explicit example

Suppose a ring $R = \mathbb{C}[x, y]$, An ideal $I = (y^2 − x^3 + x^7)$ and $M = (R/I )_I \oplus (R/I )$. We can easily check that $R/I$ is an integral domain thus $I$ is a Prime Ideal. Hence ...
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1answer
75 views

Finding $\text{Ext}_R^n(M,R)$ and $\text{Ext}_R^n(M,N)$ for a particular case

Suppose we have the ring $R=\mathbb{Z}[x_1,x_2]$ and ideals $I=(2x_2)$, $J=(x_1x_2)$. Now, consider $M:=R/I$ and $N:=R/J$. I am trying to determine the following: (1) $\text{Ext}_R^n(M,R)$ for all $n\...
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Cohomology of surfaces without homology

I'd like to find a book or other source with a detailed calculation of the cohomology of the connected sum of n torus and of the connected sum of n real proyective planes. It can assume that you ...
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1answer
22 views

What is the homological connectivity of the complete abstract simplicial complex?

I am trying to understand the concept of homological connectivity of an abstract simplicial complex. Specifically, I am trying to compute the homological connectivity of the complete abstract ...
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1answer
60 views

No map $\mathbb{C}P^n \rightarrow \mathbb{C}P^m$ inducing a nontrivial map $H^2(\mathbb{C}P^m;\mathbb{Z}) \rightarrow H^2(\mathbb{C}P^n;\mathbb{Z})$

This is part of an exercise from Hatcher's Algebraic Topology text. The exercise is as follows: Using the cup product structure, show there is no map $\mathbb{R}P^n \rightarrow \mathbb{R}P^m$ ...
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1answer
124 views

Isomorphism not natural in $X$?

$\require{AMScd}$ I am working on the following task: Let $\mathcal{H}_*$ be a homology theory and let $X \neq \emptyset$ be a space. Construct an isomorphism $\mathcal{H}_n(X) \cong \widetilde{\...
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95 views

A question about Mayer-Vietoris sequence [duplicate]

Let $A$ and $B$ be two open sets in $\mathbb{R}^n$ ($n>1$), with $A \cup B=\mathbb{R}^n$ and $A\cap B\not=\emptyset$. There is a version of the Mayer-Vietoris sequence for reduced homology that ...
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2answers
230 views

Torsion in the integral (co)homology of Eilenberg-MacLane spaces

I've had trouble finding a reference. I know that the classifying space of a group $G$ is an example of an Eilenberg-MacLane space $K(G,1)$, so that the cohomology of $K(G,1)$ is group cohomology. If ...
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85 views

Infinite (co)-homology

Lately, I've been wondering if it was possible to define singular homology also with infinite-dimensional simplices. For example we could define an infinite dimensional simplex as: $$\Delta_{\infty}:=\...
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1answer
62 views

Linear action induced in homology of the $n$-dimensional torus

I'm reading a lecture notes on actions of $\mathbb{Z}^{p}$ on the torus but I think that are not selfcontained. I need to know the definition of the "induced action in homology" of a given ...
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25 views

Long exact sequence of a graph $(G, V)$

In my algebraic topology course, we are studying homology groups of graphs. Somewhere in a proof, we used the fact that the long exact sequence of the graph $(G, V)$ (where $G$ is a Hausdroff space, ...
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2answers
57 views

For a graph $(G, V)$, $\# \text{edges}$ is given by the rank of $H_1(G, V)$.

In my algebraic topology course, we are studying homology groups of graphs. We showed that for $(G, V)$ a graph (where $G$ is a Hausdroff space, the edges, and $V \subset G$ is finite subset, the ...
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0answers
38 views

Topological properties of the chord space $C^2(X)$ and it’s generalizations

The chord space of $X$ is defined as the set $$C^2(X)=\frac{X\times X}{(x_1,x_2)\sim (x_2,x_1)}$$ In general, $C^n(X)$ is the set of unordered $n$-tuples with points in $X$: $$C^n(X)=\frac{X^n}{x\sim\...
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50 views

Sheaf homology defined in terms of Tor

By the general philosophy of cohomology, cohomology is essentially derived $\operatorname{Hom}$ (i.e. $\operatorname{Ext}$), and homology should be derived tensor product (i.e. $\operatorname{Tor}$). ...
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1answer
47 views

Singular homology with coefficients in a ring versus in an abelian group

As described here and here the singular homology of a topological space $X$ with coefficients in a ring $R$ is given by a bunch of $R$-modules $H_n(X,R)$. However, sometimes I see people talking about ...
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47 views

Equivalence of homology theories via acyclic models

I'm looking for a proof of "any two homology theories are equivalent" (obviously with some other hypothesis) via the Acyclic Models Theorem. I know that this is an application of the Acyclic ...
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0answers
71 views

Step in proof on the support of a module in equivariant cohomology

I've been stuck on one step of a proof, and am hoping someone will have a helpful hint/explanation :) Here is the setting:we have two Lie groups $K \hookrightarrow T$ acting on a manifold $M$. We can ...
3
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2answers
78 views

Coefficients in homology

The singular homology of a space $X$ is defined to be the homology of the chain complex $${\displaystyle \ldots {\stackrel {}{\longrightarrow }}\mathbb Z[Sing_2(X)]{\stackrel {}{\longrightarrow }}\...

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