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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1answer
16 views

Definition of the Fundamental Class for $K(A,0)$

I'm having a little doubts on the definition of the fundamental class for the Eilenberg-MacLane space $K(A,0)$. Recall that a fundamental class $\imath_{A,n}$ for a polarized $K(A,n)$ is the element ...
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1answer
16 views

$f_*$ induces an isomorphism in homology iff the mapping cone of $f_*$ is contractible.

Let $f_*:C_*\to D_*$ be a chain map. I'm stuck in the proof of the following statement: $f_*$ induces an isomorphism in homology iff the mapping cone of $f_*$, cone($f_*$), is contractible. (For ...
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0answers
28 views

Injectivity in the zero homology

I'm struggling with following step in an excercises about Mayer-Vietoris sequences: In one step the solution says this map is injective since $A \cap B$ is path-connected: $$ H_0(A \cap B) ...
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0answers
18 views

Long exact sequence of $(I \times Y, \partial I \times Y)$.

There's a section in chapter 3 in Hatcher's Algebraic Topology where he talks about the long exact sequence of the pair $(Y \times I, Y \times \partial I)$, where $Y$ is any topological space. The ...
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1answer
39 views

How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [on hold]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ ...
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0answers
20 views

universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
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1answer
43 views

Can binary ring for homology make life easier?

Do you know of a proof which uses homology to demonstrate a property about a topological space which is made easier (or even possible) because they work over $\mathbb{Z}/2\mathbb{Z}$ instead of ...
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0answers
31 views

$H^*(X,A;R)\cong H^*(X',A';R)\; \Rightarrow H^*(X\times Y ,A\times Y;R)\cong H^*(X'\times Y,A'\times Y;R)?$

I have a quastion about product spaces in singular cohomology. I only know a formula for sinugular homology for product spaces from lecture, the universal coefficient theorem. Let $R$ be a ...
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0answers
26 views

Simple modules and homological algebra

Let $A$ be an k-algebra, and $M$ a $A-$module. If $Ext^{1}(M,S)=0$ for every simple $A-$module $S$, then $M$ is projective. I know that this is true if $A$ is finite-dimensinal, but if $A$ is ...
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1answer
35 views

If induced map on homology is surjective, is induced map on cohomology injective?

Suppose I have topological spaces $X, Y$ and a continuous map $f: X \to Y$. Let $\mathbb{k}$ be a field, and $i \ge 1$ an integer. If the induced linear map on homology $f_* : H_i ( X, \mathbb{k}) ...
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0answers
20 views

When does the first cohomology group commute with inverse limit?

Let $M_i,i\in\mathbb{N}$ be an inverse system of continous, discrete G-modules and let $M=\varprojlim M_i$. Under what conditions on $M$ and $M_i$ do we have $\varprojlim H^1(G, M_i) \cong H^1(G, M)$? ...
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1answer
40 views

Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class ...
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1answer
40 views

$T^2-D$ does not retract to the boundary $\partial D$

First of all: yes, there is already a post about it, but I missread retract as strong deformation retract and wanted to know if this solution is right if we really do assume the stronger assumption of ...
1
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1answer
42 views

Why is $H^*(S^1\times X,\{\text{pt}\}\times X;R)\cong H^*(D^1\times X,\partial D^1\times X;R)$?

Let $X$ be a topological space, $R$ is a commutative, unital ring. In a proof from lecture there is claimed that $H^*(S^1\times X,\{\text{pt}\}\times X;R)\cong H^*(D^1\times X,\partial D^1\times X;R)$ ...
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0answers
32 views

Poincaré duality isomorphism maps cohomology to homology here?

Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i: M \to A$. Let $k = p - n$. Does the Poincaré duality isomorphism$$\bigcap \mu_A: H^k(A) \to H_n(A)$$map the ...
1
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1answer
20 views

Do these non-homotopic maps induce the same map in reduced homology?

Consider two maps $f, g: X\to Y$, where $X=Y=\{ 0, 1 \}$ with discrete topology, $f$ is the identity and $g$ maps everything to 0. Then it's clear that $\widetilde{H}_0(X;\mathbb{Z})\cong \mathbb{Z}$ ...
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1answer
12 views

On the assumptions of cocyle conditions in a Lie algebra

To define the Cohomology (with values in $\mathbb{C}$) on a lie algebra $L$, we define a coboundary map $\delta:\Lambda^n(L)\to \Lambda^{n+1}(L)$. There is a general formula for the coboundary map but ...
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0answers
27 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
2
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0answers
23 views

Cohomology of geometric realization of a simplicial topological space

Let $X$ be a simplicial topological space. We can consider to notions of cohomology of $X$. Denote by $|X|$ geometric realization of $X$. Then we can take just $H^*(|X| )$ (i.e. usual cohomology of ...
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1answer
36 views

A remark about the map $\partial^*:H^{*}(A;V)\to H^{*+1}(X,A)$ of the l.e.S. in cohomology

My question is about a remark from lecture about the connecting-homomorphism of the long exact sequence in homology of a pair $(X,A)$. Let $(X,A)$ a pair of topological spaces, $V$ be an abelian ...
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1answer
45 views

counterexample for $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence

I want to understand a counterexample for: Let $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence, because the statement ...
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0answers
43 views

Characterisation of Stable Cohomological Operations: $\Sigma (\tau_n (\imath_{A,n}))=\rho_{n+1}^*(\tau_{n+1}(\imath_{A,n+1}))$

I've started studying (stable) cohomological operations on my lecture notes, and I was given that an equivalent definition for a family of cohomological operations to be a stable cohomological ...
3
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1answer
39 views

Absolute Galois group $\text{Gal}(\overline{K}/K)$ of any number field $K$ has a non-open subgroup of any prime index $p$?

Let $K$ be a number field, and let $p$ be a prime number. Does $G = \text{Gal}(\overline{K}/K)$ necessarily have a subgroup of index $p$ that is not open?
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0answers
66 views

Showing that the pull back of a volume form under inclusion Is the generator of the relative cohomology group $H^{n}(\mathbb{R}^n,\mathbb{R}^n-0)$

sorry if this is a bit low level. I have to learn a lot about cohomology from scratch very, very quickly so I'm having a hard time keeping the ideas in my head. As the definition of relative ...
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0answers
47 views

Cohomology algebra generated by Steifel-Whitney classes and dual classes subject to defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
3
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1answer
28 views

Left vs right projective resolutions and homology of monoids

Let me use the ad hoc notation $\mathbb Z^l$ and $\mathbb Z^r$ to distinguish between left and right modules. These are trivial modules. The homology of a (discrete) finite monoid $M$ with ...
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0answers
39 views

Killing vector field for Witten complex?

I am reading a classical paper by Atiyah Bott "The moment map and equivariant cohomology". In paragraph "Relation with Witten complex" (at the very beginning of this paragraph) they claims that ...
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1answer
22 views

Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold

The symplectic blow-up of a compact symplectic manifold $(X,\omega)$ along a compact symplectically embedded submanifold $(M,\sigma)$ results in another compact manifold $(\tilde{X},\tilde{\omega})$ ...
3
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1answer
61 views

Compatibility between Unreduced Suspension Iso and Reduced Suspension Iso

I need some clarifications on these two "basic" things because I realised I was using them carelessly and now I want to know once and for all the relation between the two. Let us assume working with ...
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0answers
23 views

How to find a counter-example that the centralizer of differential graded algebras does not preserve quasi-isomorphism?

Let $A^{\bullet}$, $B^{\bullet}$ be two differential graded algebras (dga) and $f: A^{\bullet}\to B^{\bullet}$ be a differential graded homomorphism between them. Now let $R$ be another algebra ($R$ ...
4
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1answer
46 views

Homology with local coefficients as a functor from pointed, path-connected spaces and $\pi_1$-modules.

A local system of coefficients on a space $X$ is a functor $F\colon \Pi(X)\rightarrow Ab$ from the fundamental groupoid to the category of abelian groups. From this, one can define the homology groups ...
2
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1answer
49 views

$n \times n$ invertible matrix defining diffeomorphism

I was reading the proof of the Hairy Ball Theorem in Madsen and Tornehave's book "From calculus to cohomology", and at some point they refer to the Lemma 6.14, which says the following: An invertible ...
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0answers
25 views

Relation between compactly supported cohomology and locally finite homology

Building up on a previous question, I am currently investigating in the properties of locally finite homology. Suppose that $X$ is a reasonably well-behaved space. I want to find out whether there is ...
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1answer
40 views

Basic question on cohomology ring

To show (1) $S^2\vee S^1\vee S^1$ is not homotopy equivalent to $S^1\times S^1$ (2) $S^1\vee S^2\vee S^3$ is not homotopy equivalent to $S^1\times S^2$ I use the same method: For (1) the ...
3
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1answer
32 views

On the boundary map of a locally finite chain complex

I am just learning about locally finite homology and I'm having a bit trouble understanding some of its concepts. There doesn't seem to be a whole lot of (non-advanced) literature on this topic, so I ...
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2answers
42 views

Is there any expression to calculate the homology groups of a quotient space?

Let $B \subset A$ where $A$ is a topological space and $A/B$ the space obtained from $A$ via collapsing $B$ to a single point. I was wondering if there is any expression for $H_k(A/B)$ in terms of ...
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0answers
27 views

Homology of SO(3)

In Tohru Eguchi, Peter B. Gilkey, and Andrew J. Hanson's review paper "Gravitation, gauge theory and differential geometry," I came across the following claim about the Homology of SO(3): I cannot ...
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1answer
35 views

Cohomology of Eilenberg Maclane space

In a book on spectral sequences that I am reading, it is stated, without proof, that $H^i(K(\mathbb{Z},2);H^0(K(\mathbb{Z},1);\mathbb{Z}))$ is isomorphic to $\mathbb{Z}$ for even $i$ and $0$ for odd ...
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4answers
1k views

Why Cohomology Groups?

Why do we need cohomology groups? Homology groups are easier to compute and given two topological spaces, there is an isomorphism in homology groups if and only if there is an isomorphism in ...
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1answer
44 views

Is Hatcher's proof of thom isomorphism theorem flawed?: I don't believe that $H^n(E,E_0)\cong H^n(R^n,R^n-0)$

Let $E$ be an oriented vector bundle over $B$, a CW complex, with fiber of dimension $n$. Let $E_0$ be $E - B\times 0$. The main theorem Hatcher uses to prove the thom isomorphism theorem is that the ...
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1answer
34 views

Can the cohomology of the Grassmannian identified with the cohomology of a specific dense open subvariety?

Let $(\mathbb{C}^{2p},Q)$ be a $2p$-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form $Q$ where $p\geq 3$. Let $l\leq p-2$. You may assume that $l$ is odd if this ...
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2answers
34 views

On the surjectivity of the Hurewicz homomorphism

The Hurewicz homomorphism is a surjective homomorphism from $\pi_n(X) \to H_n(X)$ if $\pi_{n-2}(X)=0$ according to Wikipedia. But if it is surjective then how could the following (contradiction) I ...
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0answers
7 views

notation in mosher tangora: What is $C(\sigma \otimes \sigma)$?

On page 16 of mosher and tangora, where they are talking about carriers and the cup-i product, they write $C(\sigma \otimes \sigma)$. What does $C(\alpha)$ where $\alpha$ is an $n-cell$ in $C_n(X)$ ...
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0answers
19 views

stable cup-i product

I am confused on the definition of the cup-i product in page 15 Mosher and Tangora's cohomology operations and applications in homotopy theory. Let $X$ be a simplicial complex and $S^\infty$ be the ...
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0answers
37 views

Can motivic cohomology say anything about real points?

Let $X$ and $Y$ be two smooth schemes defined over $\mathbf Z$. Suppose that we have a scheme morphism inducing isomorphisms on motivic cohomology groups $\mathrm H^p(Y,\mathbf Z(q)) \simeq \mathrm ...
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2answers
96 views

relations between homology and cohomology

Let $p$ be a prime number and $X$ a topological space. Are the following equivalent? (1) In the homology module $H_*(X;\mathbb{Z})$ there does not exist any element of order $p$. (2) In the ...
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1answer
38 views

Homology of Hom and Hom of homology

In 'Homological Algebra' by Cartan & Eilenberg: (page 203) For complexes $X$ and $Y$, consider the map $\alpha':H^{p+q}(\text{Hom}(X,Y))\rightarrow \text{Hom}(H_p(X),H^q(Y))$. Let $h_1\in ...
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1answer
25 views

Product of nullhomologous curve with $S^{1}$ factor is still nullhomologous

Let $N$ be a nullhomologous curve in a $3$-manifold $X$. Let $S^1\times X$ be the product manifold. Why is then $S^1\times N$ nullhomologous in $S^1\times X$? PS: Is nullhomologous just a statement ...
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2answers
53 views

How can I calculate the homology group of an infinite torus using Mayer-Vietoris?

I want to calculate the (simplicial) homology of the following space using Mayer-Vietoris: I have tried to do it by cutting it along the axis and getting two subspaces homeomorphic to something ...
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0answers
29 views

How to prove $M\otimes_{\,G}A≅ {Hom}_{G}({Hom}_{\,\mathbb Z}(M,{\,\mathbb Z}),A)$?

Given that G is a finite group, M is a finitely generated right free G-module and A is a left G-module, there exists a natural G-isomorphism $\phi\ : M\otimes_{\,G}A\rightarrow ...