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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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Chern Classes Identification

I have a question about a remark/observation in A Concise Course in Algebraic Topology by P. May at page 199. Here the excerpt: Having introduced the Chern classes May claims that $c_1 \in H^2(BU(...
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Problem on simplicial complexes.

If $(S_0,P_0)$ and $(S_1,P_1)$ are abstract simplicial complexes, a simplicial map from $(S_0,P_0)$ to $(S_1,P_1)$ is a function $f\colon S_0 \longrightarrow S_1$ such that, if $U\in P_0$, then $f(...
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1answer
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Hatcher Exercise 3.2.16

Hatcehr Exercise 3.2.16. Show that if $X$ and $Y$ are finite CW complexes such that $H^∗(X;\mathbb Z)$ and $H^∗(Y;\mathbb Z)$ contain no elements of order a power of a given prime $p$, then the ...
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Construction of the Stiefel-Whitney Classes

I have a question about an argument in a proof from J.P. May's A Concise Course in Algebraic Topology (page 196): Question: I don't understand what is meant by interpreting the Steenrod operations "...
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ranks $dim(C_{d}(K), dim(B_{d-1}(K)),dim(Z_{d}(K))$ for K a simplex of n vertices

given n vertices (0-dimensional simplices) consider the full simplex K generated by these n vertices. choose a dimension $d \leq n-1$ I need to prove the closed form formula for the following ranks ...
2
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1answer
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Understanding relative cohomology of the interval

I want to understand relative cohomology. The easiest example I could think of was $H^*(I,\partial I;G)$ where $I=[0,1]$ and $G$ is an abelian group. I know the following facts: $H^*(I,\partial I)$ ...
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First Cohomology of Abelian Cover

Let $S$ be a closed oriented surface. Consider the (universal) abelian cover $p \colon S_{ab} \rightarrow S$, i.e. the one whose group of deck transformations is the abelianization of $\pi_1(S,\star)$....
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2answers
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Is there a quick way to distinguish between a wedge of spheres and a suspended projective space?

I remember reading about the following example a while back in one of Steenrod's papers on cohomology operations. If we look at $S^3\vee S^5$ and $\Sigma \mathbb{C} P^2$, these spaces have isomorphic ...
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Understanding the Yoneda product defined in terms of morphisms of projective resolutions.

On the wikipedia page for the Ext functor, they say that one can equip the graded abelian group $\operatorname{Ext}^*:=\bigoplus_{i=0}^{\infty}\operatorname{Ext}^i(A,A)$ with the structure of a ring (...
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1answer
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What is the difference between the cohomology rings of odd and even dimensional spheres?

I am reading Hatcher example 3.16 (3.18 in the old edition). It says that the cohomology ring of an odd-dimensional sphere is $\Lambda_\mathbb{Z}[\alpha]$ which is the exterior algebra generated by ...
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$H^2(G,A)$ is in bijection with the class of extensions of $A$ by $G$ - does this depend on the action of $G$ on $A$?

Let $G$ be a profinite group, and $A$ an Abelian group. Given an extension of $A$ by $G$, $0 \longrightarrow A \longrightarrow E \longrightarrow G \longrightarrow 1$, it is known that we can make $A$...
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CoRes $\circ$ Res $ = [G:H]Id$ on the cohomology groups of a profinite group

Let $G$ be a profinite group, $H \leq G$ open. It is known that thus $H$ is closed and has finite index in $G$. Any $G-$module is an $H-$module and one can construct the restriction map as the ...
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Singular Homology - Calculate The Boundary of This n- Simplex.

Let X be a topological space. Define the singular chain complex $C(X)$ ,with the group of n-chains $C_n(X)$, equal the free abelian group generated by singular n-simplices:$\sigma:\Delta^n \rightarrow ...
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Cohomology generalized Quaternions

Good day to everyone I have a doubt about where I can find about the cohomology of the generalized Quaternions. I managed to find something on the book "Homological Algebra" by Henri Cartan and ...
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Cohomology theories and sheaf cohomology

Let $X$ be a paracompact Hausdorff topological space, $\mathcal U$ an open covering of $X$ and $\mathcal N(\mathcal U)$ the nerve of the covering (https://en.wikipedia.org/wiki/Nerve_of_a_covering). ...
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Modifying long exact sequences

Let $$\dots A_i\stackrel {f_i}\to B_i \stackrel {g_i}\to C_i \stackrel {h_i}\to A_{i+1}\to \dots$$ be a long exact sequence of Abelian groups. Is it true that if there are maps $k_i:D_i\to E_i$ such ...
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Confusion regarding sheaf cohomology and singular cohomolgy

Let $X$ be a smooth, projective curve (in particular irreducible) of genus $g$ at least $1$. We know that $H^1_{\mbox{sing}}(X,\mathbb{Z})=2g$. But, since $X$ is irreducible, the locally constant ...
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24 views

Lemma for Hurewicz Theorem (Bredon)

I am trying to understand the following lemma: If $f,g:I\rightarrow X$ are paths s.t. $f(1)=g(0)$ then the 1-chain $f*g-f-g$ is a boudary. Proof: On the standard 2-complex (should it say simplex?) $...
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1answer
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homology of a subspaces of product of circles.

I do not how to approach the computation of the homology of the following space $X = \{ (z_1, z_2, z_3) \in (S^1)^2 \times D_2 : z_1 + z_2 + z_3 = 0 \}$ $S^1$ denotes the unit circle and $D_2$ the ...
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1answer
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Is every 3-dimensional Poincaré complex a 3-dimensional topological variety?

I have this question, if every 3-dimensional Poincaré complex is a 3-dimensional topological manifold? Definition (Poincaré complex) $X$ is a n-dimensional Poincaré complex if $X$ have the same ...
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Homology of the Torus over a Module

The homology groups $H_n(T,\mathbb{Z})$ for the torus $T$ are $\mathbb{Z} \oplus \mathbb{Z}$ for $n=1$, $\mathbb{Z}$ for $n=0,2$. Zero otherwise. Question: Changing the coefficients in $\mathbb{...
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Where is the inclusion map being used in the proof of Corollary 2.24 from Hatcher's AT? [duplicate]

Where is the inclusion map being used here? How is proposition 2.22 being used? What is wrong with the following proof: Since $B/(A\cap B) \cong (A\cup B)/A = X/A$, then $\tilde H_n(B/(A\cap B) \...
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Cohomology of spectra as free modules over Steenrod Algebra

I'm studying the construction of Adams spectral sequence in Hatcher's "Algebraic Topology" chapter five; I'm in trouble with an "algebraic statement". Let $X$ be a connective CW-spectrum of finite ...
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1answer
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What is the tensor product over in Künneth’s formula

Let $X$ and $Y$ be CW complexes. Fix a ring $R$. Künneth formula says $$ H^k(X\times Y, R)\cong \bigoplus_{r+s=k} H^r(X,R)\otimes H^s(Y,R)$$ I am not able to see a reference where it is mentioned ...
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Compute Cech cohomology of a subscheme defined by a homogeneous equations in $P^2_k$

I am trying to compute the cohomology in Hartshorens's exercise 4.7 in chapter3. Given a subscheme $Y$ in $X=P^2_k$ defined by a homogeneous equations $f(x_0,x_1,x_2)$ of degree $d$, and point (1,0,0) ...
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2answers
63 views

Why is $H_0(X,x_0) \ne H_0(X)$

We know $\tilde H_n(X) \cong \tilde H_n(X,x_0)=H_n(X,x_0)$ for all $n\ge 0$. Also, when $n>0$, we have $H_n(X)\cong \tilde H_n(X) \cong H_n(X,x_0)$. However, when $n=0$, we have $H_0(X)\cong \...
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0answers
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How is $\tilde H_i(S^n) =0$ for $i \ne n$?

I understand the proof of Corollary 2.14 and I know how to show $\tilde H_n(S^n) \cong \mathbb Z$ from the propositions 2.6 and 2.8. However, I don't see how to get $\tilde H_i(S^n) =0$ for $i \ne n$....
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How to construct birelative Hochschild homology?

I am working through Lodays book "Cyclic Homology". For an unital $k$-Algebra $A $ ($k$ being some ring) and a two-sided ideal $I$ of $A$ he defines the relative Hochschild homology $HH_\ast(A,I)$ as ...
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3answers
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First and third homology of $S^5/Z_q$ and Leray spectral sequence

I read from an article that the space $X=S^5/Z_q$ is not a Lens space because the orbifold action is not compatible with the action of the Hopf fibration $S^1\longrightarrow S^5\longrightarrow CP^2$. ...
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1answer
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Can someone please explain the map $S:C_n(X) \to C_n(X)$ from Hatcher's AT page 122?

Barycentric Subdivision of General Chains. Define $S:C_n(X) \to C_n(X)$ by setting $S\sigma = \sigma_\# S\Delta^n$ for a singular $n$-simplex $\sigma: \Delta^n \to X$. How are we using $S$ in the ...
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group cohomology with coefficients in a complex

I am reading Brown's "cohomology of groups" when he introduces the group homology and cohomology with coefficients in a chain complex $C_*$. (pp 168) . The homology is defined as $H_*(G, C_*) = H(...
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Homology and Cohomology of Schubert varieties

Let $X$ be a Schubert variety , seen as a sub variety of the projective variety of complete flags (over the field $\mathbb R$ or $\mathbb C$) . Is it true that $H_n(X,\mathbb Z)\cong H^n(X,\mathbb Z)...
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$H_n(X, F_p) = 0$ implies $H_n (X,Q)=0$ for certain CW-complexes $X$

Consider a CW-complex $X$ with exactly one cell in each dimension. Suppose there is a prime $p$ such that $H_n(X, F_p) = 0$ for all $n\geq 1$. How can I show that this implies $H_n (X,Q)=0$ for all ...
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Cohomology of colimit is limit of cohomology ? (group cohomology)

In Homotopy theoretic methods in group cohomology, Henn's part, section 1.2, the example following definition 1 has the following sentence "the cohomology $H^*(G,\mathbb{F}_p)$ of a group $G$, which ...
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2answers
40 views

Number of connected components Invariant

Why is the number of connected components invariant under homeomorphisms? I know that connectedness, as well as path connectedness, are properties conserved through homeomorphisms. But why is this ...
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Cohomology of Projective Space Product

What are the singular cohomology groups of $(\mathbb RP^1)^n$ and what is the cup product structure?
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Show that $\tilde{H}_i(S^n-X)\approx \tilde{H}_{n-i-1}(X)$ when $X$ is homeomorphic to a finite connected graph

Problem 2.B.2 in Hatcher's Algebraic Topology. Also the text uses $\approx$ for an isomorphism. Show that $\tilde{H}_i(S^n-X)\approx \tilde{H}_{n-i-1}(X)$ when $X$ is homeomorphic to a finite ...
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(Solved) Orientability of manifold with point removed (Hatcher exc. 3.3.2)

EDIT: I misunderstood the exercise, as pointed out by @Arthur in the comments. I only showed that a manifold does not lose orientability when a point is removed. But we can imagine a one-dimensional ...
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Modify the Alexander horned sphere for an embedding $S^2 \hookrightarrow ֓\mathbb{R}^3$ s.t. neither component of $\mathbb{R}^3 − S^2$ is 1-connected.

Question 2.B.6 in Allen Hatcher's Algebraic Topology page 176: Modify the construction of the Alexander horned sphere to produce an embedding $S^2 \hookrightarrow ֓\mathbb{R}^3$ for which neither ...
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1answer
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Concluding that $\mathbb{R}P^2$ cannot be embedded in $S^3$

I want to deduce that the projective plane $\mathbb{R}P^2$ cannot be embedded in $S^3$ using homology. My idea is to compute $H_*(S^3-h(\mathbb{R}P^2))$ for some arbitrary embedding $h:\mathbb{R}P^2\...
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1answer
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Finding a mistake using Mayer-Vietoris

I was computing the homology of $S^3-\coprod_{i=1}^4 I_i$, where $I_i=[0,1]$ for all $i$ (they are being identified with an embedding). Intuitively, this should be homotopy equivalent to $S^1$, since ...
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About $O_X$-modules?

Consider $F$ an $O_X$-module is true in general that $Hom_{O_X}(O_X,F) \equiv F(X)$ ? Morover I need to prove that if $F$ is a flasque sheaf then $H^n(X,F)=0$ for $n>0$. I think is beacuse the ...
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1answer
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Determine the homomorphism $i_*:H_1(S^3-g(M))\to H_1(S^3-g(\partial M))$.

Let $M$ be the Möbius band and $\partial M$ its boundary. Consider an embedding $g:M\to S^3$. I have to determine the homomorphism $i_*:H_1(S^3-g(M))\to H_1(S^3-g(\partial M))$ induced by the ...
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1answer
45 views

Computing homology of complement of an embedding

Let $I=[0,1]$ and $S^3$ the $3$-sphere. Assume we have injective maps $f_1,f_2:I\to S^3$ such that $\mathrm{Im}f_1\cap\mathrm{Im}f_2=\emptyset$. I have the following problem: Compute $H_*(S^3-\...
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1answer
48 views

Computing homology group

Let $X=\{(z_1,z_2) : z_1 , z_2 \in \mathbb{C}$ $and$ $ z_1 \neq z_2\}$. Compute the homology group of $X$. $\\$ My try: The space is nothing but $\mathbb{C} \times \mathbb{C} $ subtracting the ...
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1answer
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An injection yields an isomorphism in cohomology? $\Bbb RP^n \rightarrow \Bbb RP^{\infty}$.

In page 79 line 10 Hatcher made the claim that An embedding $\Bbb RP^{n-1} \rightarrow \Bbb RP^\infty$ induces an isomoprhism $$H^i(\Bbb RP^n, \Bbb Z_2) \leftarrow H^i(\Bbb RP^\infty, \Bbb Z_2)...
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0answers
27 views

Why is the Bockstein morphism a derivation?

I'm trying to understand the Bockstein morphism in cohomology, and one of the points is that $\delta : H^*(G,\mathbb{F}_p)\to H^*(G,\mathbb{F}_p)$ is a derivation that squares to $0$. I could ...
2
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1answer
46 views

Computing homology maps with non-integer coefficients

I want to compute, for example, the homology of a genus-$g$ orientable surface $M_g$ with $R$ coefficients, where $R$ is any associative, commutative, and unital ring. The construction of the surfaces ...
2
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0answers
27 views

Group cohomology and singular cohomology of classifying space

Let $G$ be a finite group, and denote by $BG = K(G,1)$ the classifying space. For any fibration $X \rightarrow E \xrightarrow{\pi} BG$, the Serre spectral sequence $E_2^{p,q} = H^p(BG;H^q(X))$ ...
2
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2answers
91 views

If $A$ and $B$ are homeomorphic proper closed subsets of $\mathbb{R}^n$, do their complements have the same homology?

Let $f_1,f_2:[0,1]\to S^3$, $g:M\to S^3$ and $h:\mathbb{R}P^2\to S^3$ be inyective maps, where $M$ is the Möbius strip. Assume that $\mathrm{Im}f_1\cap \mathrm{Im}f_2=\emptyset$. I want to compute $...