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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32 views

How we show that the cohomology rings of product of 2-sphere and connect sum of projective plane are isomorphic?

How we show that the following ring isomorphism $$H^{*}(S^{2}\times S^{2};\mathbb{Q})\cong H^{*}(\mathbb{C}P^{2}\sharp \mathbb{C}P^{2};\mathbb{Q}).$$
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Meaning of “cohomology” in algebraic geometry?

I want to learn about different cohomologies in algebraic geometry with a view towards understanding the idea of motives. I'm more interested in starting with a general overview or history. The main ...
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Computing $H_1(K)$ and $H_2(K)$, where $K$ is the complex consisting of the proper faces of a 3-simplex

I am interested in computing $H_1(K)$ and $H_2(K)$, where $K$ is the complex consisting of the proper faces of a 3-simplex. However, specifically, I am interested in doing so, if possible, by using ...
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Question about the map $S^1\to S^1$ in the context of the real projective plane $\mathbb{R}P^2$

I was recently working on an exercise to compute $H^*(\mathbb{R}P^2\times \mathbb{R}P^2,\mathbb{Z})$ and there was a particular step in the solution for which i would like to get a better intuition. ...
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1answer
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Is there a general notion of nonabelian cech cohomology?

Diving into the theory of vector bundles and characteristic classes, I've often come across the introduction of $\check{H}^1(\mathcal U, Gl(n))$, where $\mathcal U = \{U_i\}_i$ is an open cover of a ...
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3answers
364 views

“Short exact sequences”, longer than classical one

A standard short exact sequence is a complex $$0\to A\to B\to C \to 0$$ Based on this concept we get a lot of new concept in homological algebras. For example left exactness, right exactness, derived ...
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1answer
27 views

Why is $H^n(\mathbb{R}P^2;\mathbb{Z}) \otimes H^n(S^1;\mathbb{Z}) \not =0$ in degree $n = 1,2$?

Compute $$H^*(\mathbb{R}P^2\times \mathbb{R}P^2, \mathbb{Z}).$$ Hint: Consider the Mayer-Vietoris sequence of suitable open sets of the form $\mathbb{R}P^2\times U$ and $\mathbb{R}P^2\times V$ where $...
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Cutting along a non-separating curve reduce the rank of first-homology of compact surface by one

I have a problem with understanding the following lines, called the Hierarchy of Compact Surface. Let $M$ be a compact surface with non-empty boundary. Then $M$ is either closed disc $\Bbb D^2$ or ...
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2answers
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Degree of a map $T^2\to T^2$ induced by an $2\times 2$ integer matrix

Note that the $2$-torus $T^2$ can be seen as a quotient space $\Bbb R^2/\Bbb Z^2$ of $\Bbb R^2$. Then any $2\times 2$ integer matrix $A=(\begin{smallmatrix} a & b\\ c & d \end{smallmatrix})...
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1answer
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Additive functors from the category of free abelian groups are right adjoints?

I am reading Lectures on algebraic topology by Albrecht Dold. Let $t$ be an additive functor from the category of free abelian groups to the category of abelian groups. In VI 7.3 he writes: For any ...
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Fibre bundles and good pairs

Consider a fibre bundle $E\rightarrow B$. I was wondering when we can say that the pair $(E,B)$ is a good pair? By a good pair $(A,X)$ I mean two spaces $X\subset A$ such that $X$ is closed and is a ...
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An algebraic topology problem about Mayer Vietoris [closed]

I don't know how to use the Mayer-Vietoris sequence to find the homology groups of the torus and the Klein bottle. Can somebody help? Thanks a lot.
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Finding a cocycle representative of the generator of $H^n(S^n)$ which squares to zero

Let $n\geq 1$ be a natural number and let $C^n(S^n)$ be the $n$-th singular cochain group (with integer coefficients) of the n-Sphere. Let $\chi\in H^n(S^n)\cong \mathbb Z $ be the generator of the $n$...
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Understanding basics on spectral sequences for a stratified space with respect to compactly supported cohomology

I am new to the topic of spectral sequences, and I would like to understand a point in the very first page of Petersen's paper "A spectral sequence for stratified spaces [...]", whose arxiv ...
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2answers
62 views

Which 2-chain has this 1-chain as boundary?

Define $\sigma:[0,1] \to \mathrm{R}^2$ by $t \mapsto (t,0)$. And define $\sigma':[0,1] \to \mathrm{R}^2$ by $t\mapsto (1-t,0)$. Now I want to figure out which 2-chain in $C_2(\mathrm{R}^2)$ has $\...
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1answer
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Why are cochain complexes in general not quasi-isomorphic to their cohomology as differential graded algebras?

Let $X$ be a topological Space and let $C^*(X)$ denote its singular cochain complex. Since the cup-product of two cocyles is again a cocycle, we get a induced product structure on the complex of ...
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On the proof of a result of Bayer and Stillman

I'm reading through the paper A criterion for detecting m-regularity of Bayer and Stillmann and came across a proof, where I don't understand an implication. The following things may need to be ...
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2answers
45 views

How to show that $S^n$ can't embed $\mathbb R^n$

There are some hints that we can prove it by the homology group of $S^n\setminus S^k$.
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1answer
150 views

What is the structure behind $ \partial \partial M = \varnothing $?

In my lectures about manifolds, I learned about the statement $ \partial \partial M = \varnothing $ where $M$ notates a manifold and $\partial M$ its boundary. My professor said, it is similar to the ...
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2answers
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Computing homology groups of complements of $S^2$

I have to compute the homology groups of $X=\mathbb{R}^3-S^2$ and $Y=\mathbb{R}^4-S^2$. In the first case I thought that, since $X$ is not connected, its homology groups are sum of the two connected ...
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Modular form and cohomology

What is the relationship between modular forms (and modular functions) over $\Gamma \subseteq SL_2(\mathbb Z)$, modular curves $Y(\Gamma)$, its singular cohomology $H^k(Y(\Gamma), \mathbb Z)$, and the ...
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Why does morphism $P \to X$ imply that intermediate Jacobian $J(X)$ is a direct summand of $J(P)$?

I'm currently reading the sketch proof of Theorem 1 in Beavuille's notes on the Luroth problem. Let $P$ be the blow-up of $\mathbb{P}^3$ in finitely many points and smooth curves, and let $X \subset \...
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Mahowald-Hopkins theorem

I am trying to recover fully a proof of a result which is attributed to M. Hopkins in the litterature, namely the following: Let $h:S^1\to BGL_1(\mathbb S_p)$ detect the element $1-p$, then the Thom ...
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Bott&Tu, Exercise 5.12(b)

Here is a link of the book: https://math.sci.uwo.ca/~jcarlso6/DFAT.pdf I am trying to solve Exercise 5.12 (p.50), part (b). It asks: Using the Mayer-Vietoris argument prove the Kunneth formula for ...
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27 views

Co-invariants in homology of groups

Let $1 \mapsto L \mapsto G \mapsto \mathbb{Z}\mapsto 1$ be a short exact sequence such that $G$ is a group of type $FP_{\infty}(\mathbb{Q})$, $H_{1}(L;\mathbb{Q})$ is infinite dimensional and let $t\...
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28 views

Cohomology of real projective plane $RP^2$ by Mayer-Vietoris principle (BottTu book, Exercise 9.4)

This exercise asks the Čech cohomology of real projective plane $\mathbb{RP}^2$. by the good cover of the projective plane $\mathbb{RP}^2$ https://images.app.goo.gl/J5WDENSrp2AkxsHp7 There is no ...
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42 views

What is the relation between homotopy groups and homology?

It's pretty much all said in the title, I don't need very specific answers, but I'd like to know the relations between those two properties (for example, if we know the homology of a topological space,...
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85 views

(Co)homology in fluid dynamics

This is a pretty broad, vague question and may not be appropriate (apologies, feel free to close if so.) I'm going into computational fluid mechanics, but coming from a pure math and physics ...
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1answer
84 views

Fundamentals of Tor Functor for an Intro Algebraic Topology Course

I'm learning about the universal coefficient theorem in my first-semester algebraic topology course, and to state and prove the theorem we needed to introduce the Tor functor. Here the class ran into ...
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Holes and generators of homology groups [duplicate]

I'm trying to better understand singular homology groups. Standard examples like spheres and tori make it seem like $H_i(X)$ is generated by a subset that is in bijection with $i$-dimensional holes in ...
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27 views

Co-H-spaces which are also Poincaré duality spaces

In this question, it is proven that any manifold $M$ which is also a Co-H space is in fact a simply-connected homology sphere. That is, $M$ is a manifold, is simply connected, and has the homology of ...
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29 views

Equivalent definitions Stiefel - Whitney / Chern - classes

There are some details I still don't understand about the definition of Stiefel - Whitney / Chern - classes. Let $\gamma_n^1$ be the tautological line bundle over the real or complex projective space $...
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24 views

Degree of covering map $S^n \to \mathbb{R}\mathrm{P}^n$ and local degree signs

This question is both about a specific problem and a soft question about these sorts of problems. I'm reading through Hatcher's textbook and having trouble dealing with signs when it comes to local ...
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Quillen cohomology of Lie algebras

Fix a base ring $k$ and $k$-Lie algebras $\mathfrak{s}$ and $\mathfrak{t}$, and consider the slice category $\mathfrak{s}/\mathrm{Lie}_k/\mathfrak{t}$. This is a category of universal algebras, so ...
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1answer
28 views

Cohomology of n-Sphere by Mayer Vietoris sequence

According to the method in https://planetmath.org/exampleofcohomologyandmayervietorissequence The Mayer-Vietoris sequence $0\rightarrow H^m(S^{n-1})\rightarrow H^{m+1}(S^n)\rightarrow0$. is obtained. ...
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1answer
50 views

Proving the Weak Continuity Property is Equivalent to the Continuity Property (in Cohomology)

In Edwin H. Spanier's Algebraic Topology, he describes both a "continuity property" and a "weak continuity property" for cohomology theories. (He actually defines these in terms of ...
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1answer
49 views

Lemma 6.1 In Munkres’ “Elements of Algebraic Topology"

In Munkres’ book, he proves Lemma 6.1 as follows: Lemma $6.1$. Let $L$ be the complex whose underlying space is a rectangle. Let $BdL$ denote the complex whose space is the boundary of the rectangle. ...
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2answers
116 views

long exact sequence of homology groups

In the long exact sequence $\cdots \to \tilde{H} (A) \to \tilde{H}(X) \to H(X, A) \to \cdots$, why isn’t the third homology in the sequence given in a reduced homology? Also I am wondering why the ...
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0answers
45 views

Cellular Homology and Arbitrary Coefficients

I'm trying to get the hang of working with homology with arbitrary coefficients. Given any abelian group $G$ and a topological space $X$, we can tensor the singular chain complex $\cdots \to C_{n+1}(X)...
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33 views

Basis of $H^\ast(F,R)$ for a fiber bundle

I was reading through the statement of the Leray-Hirsch theorem in Hatcher's AT, and I came across a condition that revealed some misunderstandings I have. In the following, $R$ is a commutative ring, ...
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1answer
57 views

Schubert calculus

Let $X = Gr(2,4)$ the complex Grassmannian of $2$-planes in $V = \Bbb C^4$ and $S$ the tautological bundle, $Q$ the quotient bundle. The cohomology ring is generated by $c_1(S), c_2(S)$ with relations ...
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2answers
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Computing the cohomology groups of the Klein bottle as a $\Delta$-complex

I am currently working on how to compute the cohomology and ring structure of certain surfaces who are given as $\Delta$-complexes such as the Kein bottle pictured below. For this i encountered this ...
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58 views

Analyzing the homomorphisms in the long exact sequence of a manifold pair

Let $M = (S^2 \times S^1) \setminus B$, where $B$ is a "small" open ball in $S^2 \times S^1$. We have the following exact sequence in homology with integer coefficients (arising from the ...
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53 views

Homology groups of $(S^2 \times S^1) \setminus B$, where $B$ is a small open ball

Let $M = (S^2 \times S^1) \setminus B$, where $B$ is a small open ball in $S^2 \times S^1$. First question: How do we compute the absolute and relative homology groups of $M$ (with integer ...
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1answer
34 views

Vanishing Relative Homotopy Groups

Let $X$ be a path-connected topological space and $A$ a subspace of $X$. Fix an integer $k$ greater than zero and suppose every map from a connected $i$ dimensional polyhedron to $X$ is homotopic to a ...
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1answer
53 views

coefficient ring for cohomology modules

I am a bit confused about a notation I am seeing while reading a paper. So basically I have a continuous map between topological spaces $f:X\to Y$, and I know that this induces a map on cohomology $f^*...
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0answers
37 views

Boundary Map of Bar Resolution vs. Face Map of the Nerve of a Group

For a discrete group $G$ I have the following two definitions, which I think are correct: The nerve of $G$ is $NG$, a simplicial set whose $n$-simplices are $G^n$ ($G^0$ being the trivial group $\{1\}...
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0answers
62 views

Group multiplications in $Spin(d)$ via $(a,q) \in (\mathbb{Z}/2, SO(d))$

We know that the short exact sequence $0 \to \mathbb{Z}/2\to Spin(d) \to SO(d) \to 1$. Given the groups $A$ and $Q$, we require to have the additional data the 2-cocycle $ f \in H^2(BQ,A)$ and the map ...
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1answer
44 views

(Co-)Homology with different coefficients [closed]

I'm recently diving into the realm of homology and cohomology and encountered the universal coefficient theorems and concluded from it $H_i(X, k) = H_i(X,\mathbb{Z}) \otimes k$ for characteristic 0 ...
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1answer
48 views

group extension properties: split, central, cocycle or not

Given a group extension $$0 \to A \to E \to G \to 1 $$ with $A$ an abelian group, there are several properties to describe this extension: central or non-central extension split or not trivial or ...

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