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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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Lens space and its generalization, homotopy/homology groups

The three-dimensional lens spaces $L(p;q)$ are quotients of $S^3$ by $\mathbb{Z}/p$-actions. More precisely, let$p$ and $q$ be coprime integers and consider $S^3$ as the unit sphere in $\mathbb C^2$. ...
3
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1answer
54 views

Injection on homology dual to surjection on cohomology?

Let $f:X\to Y$ be morphism between two topological spaces. We know that $H_k(X)\xrightarrow{f_*} H_k(Y)$ is injective if $H^k(Y)\xrightarrow{f^*} H^k(X)$ is surjective. I want to know if the ...
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0answers
49 views

Pontryagin square: Down-to-earth computer numerical values and maps

Suppose I am just a computer, and I can only read numerical values, and I cannot read any complicated math relations. Consider the most naive simple Pontryagin square, I want to translate this to a ...
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0answers
22 views

Künneth theorem for compactly supported cohomology of manifold.

I know the Künneth theorem for ordinary cohomology of a manifold. Is there any version for the compactly supported cohomology (over Rational) of manifolds. If we take the product of real projective ...
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1answer
10 views

Top non-zero Betti number of connected manifold of finite type.

The Top non-zero Betti number of a closed oriented manifold is one. is it true for the general manifold or not?
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0answers
26 views

Computing Ext for a complex of modules, help with a proof in Stacks Project

I am stuck on a step in the proof of Lemma 15.66.2 here. Let $R$ be a commutative ring with identity and let $K^{\bullet}$ be a complex of $R$-modules. I am stuck on the following sentence: "Choose a ...
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0answers
25 views

The Fubini-Study metric $ \omega_{ \mathrm{FS} } $ on the projective space $ \mathbb{P}^1 $

I have a small question about the Fubini-Study metric $ \omega_{ \mathrm{FS} } $ on the projective space $ \mathbb{P}^1 $ appearing in page : $ 119 $ of Daniel Huybrechts's book intiteled : Complex ...
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0answers
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+50

Homology in all dimensions of a handle body decomposition $D^m\cup H^{\lambda_1}\cup\dots\cup H^{\lambda_N}$ for $H^{\lambda_k}$ a $\lambda_k$-handle

Let $X_k=X_{k-1}\cup_{\theta} H^{\lambda_k}$ be the $\lambda_k$-handle attached to a $\dim X_{k-1}$-manifold along the embedding map $\theta:S^{\lambda_k-1}\times D^{\dim X_{k-1}-\lambda_k}\to\partial ...
2
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0answers
41 views

$S^2 \times S^4$ is not homotopy equivalent to $\mathbb{C}P^3$ using cohomology rings

I am trying to show that $S^2 \times S^4$ is not homotopy equivalent to $\mathbb{C}P^3$ using cohomology rings. I know that $H^*{\mathbb{C}P^3} \simeq \mathbb{Z}[\lambda]/(\lambda^4)$ as a graded ...
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0answers
42 views

Gaining an appreciation for homology class representatives in $\mathbb CP^n$.

Given a compact oriented submanifold $N \subset M$ one says that $N$ represents a homology class in $M$ by taking $i_*(\tau_N)$ where $i_*$ is induced by inclusion and $\tau_N$ is the fundamental ...
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1answer
17 views

Example of rationally acyclic open non-oriented manifold.

A manifold is rationally acyclic if its reduced cohomology is zero over rational. If the manifold is closed odd dimensional or closed oriented even dimensional then it is not possible due to Euler ...
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0answers
30 views

Retraction $S^2 \vee S^4 \rightarrow S^2$ collapsing $S^4$ to the wedge point induces homomorphism on second cohomology.

Let $r: S^2 \vee S^4 \rightarrow S^2$ be the retraction collapsing $S^4$ to the wedge point. I am trying to show that the induced map $r^{*}: H^{2}(S^2) \rightarrow H^{2}(S^2 \vee S^4)$ on the second ...
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0answers
15 views

Reduced cohomology relation to regular cohomology

I just wanted to double check that I'm not making a mistake here. For the $0$th cohomology group $H^{0}(X;G)$ of a space $X$, we can think of the elements as being functions $X \rightarrow G$ that are ...
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0answers
47 views

Question about showing $\mathbb{R}P^{3}$ is not homotopy equivalent to $\mathbb{R}P^{2} \vee S^3$.

A popular exercise is to show that $\mathbb{R}P^3$ is not homotopy equivalent to $\mathbb{R}P^2 \vee S^3$. The standard way is using cup products. This has been asked several times in various places ...
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0answers
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When compactly supported cohomology ring is zero?

The compactly supported cohomology (over rational) of a compact manifold is the same as ordinary cohomology. Also, we can relate the compactly supported cohomology of the oriented non-compact manifold ...
8
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1answer
95 views

Linking number and cup product

Let $S^p$ and $S^q$ be disjoint spheres in $\mathbb{R}^n$ with $n=p+q+1$ and let $X= \mathbb{R}^n- (S^p\cup S^q)$. By Alexander duality, their fundamental classes yield cohomology classes in $\tilde{H}...
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0answers
21 views

Thom spectrum $MSpin$ and $E_2$-page for a large degree $i\geq 8$

Let the $MTH$ is Madsen-Tillman spectrum (which is a close cousin of the more usual Thom spectrum $MH$) associated to tangential structure $H$. For a computation involving no odd torsion, the Adams ...
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1answer
26 views

The direct sum of any family of objects

Suppose in an Abelian category $\mathscr C$, the direct sum of any family of objects exists, then is $\bigoplus_{i\in\varnothing}A_i$ equal to 0 or meaningless?
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0answers
28 views

Homotopy Invariance: Prism Operator in Hatcher Pg 112, Proof of Prop 2.10

Can someone please help me understand the prism operator and how it helps form a chain homotopy? A toy example would greatly help! I don't quite understand why $F \circ (\sigma \times {I})$ is needed ...
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0answers
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Importance of cohomological connectivity of manifolds.

The cohomological connectivity of manifold M (not rationally acyclic) is defined as: The smallest positive integer m such that $H^{m}(M)$ (over rational) is not zero. I need the good references for ...
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2answers
49 views

Homotopy type of a disk and homotopy type of a sphere.

According to the following link : Calculation of de Rham complex for real projective space , $ \mathbb{P}^d $ is devided in two open sets : $ U = \{ \ [ x^0 : \dots : x^d ] \ | \ x^d \neq 0 \ \} $ and ...
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0answers
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How to compute $ H^{\bullet} (\mathbb{P}^n (\mathbb{R}), \mathbb{Q}) $, using Mayer Vietoris sequence?

How do we compute the Cohomology algebra of the real projective space: $ H^{\bullet} (\mathbb{P}^n (\mathbb{R}), \mathbb{Q}) $, by induction, and by using Mayer Vietoris sequence ? Thanks in advance ...
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1answer
38 views

What's the meaning of “decreasing filtration”?

Page 152-153, "Algebraic Geometry" by Lei Fu. The condition (e) of the definition of spectral sequence is listed as follows: (e) A family of objects $H^n(n\in \Bbb Z)$ in $\mathcal C$ and each $H^...
2
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1answer
45 views

Product structure in Cohomology Spectral sequence

In the Serre Spectral sequence, we know, the cup product structure induces a canonical product in all $E_{r}$ pages which is compatible with respect to the differential. I am trying to understand ...
3
votes
1answer
40 views

Segre Map induces Cohomology Morphisms

Consider the complex Segre map $s:\mathbb{CP}^1 \times \mathbb{CP}^1 \to \mathbb{CP}^3$ given by $([x_0:x_1],[y_0:y_1]) \mapsto [x_0 y_0: x_0 y_1: x_1 y_0: x_1 y_1]$. We know that $H^*( \mathbb{CP}^...
2
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0answers
43 views

Singular chain complex as a graded algebra

Let $X$ be a topological space and denote $S_*(X)$ the singular chain complex of $X$. There is a chain map (The Eilenberg-Zilber map) $$E: S_*(X)\otimes S_*(X) \rightarrow S_*(X \times X)$$ which ...
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0answers
14 views

Natural isomorphism for a reduced cohomology theory on CW complexes.

I've found here: http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf the following lemma. $\textbf{Lemma 2.}$ If a reduced cohomology theory $h^{*}$ defined on C complexes has $h^{n}(S^{...
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2answers
70 views

How to prove the sufficient and necessary conditions of pullback square?

Let $\mathscr C$ be an Abelian category and $W,X,Y,Z$ objects in $\mathscr C$. How to prove the following lemma? Lemma: The square $$ \require{AMScd}\begin{CD} W @>u >> X \\ @VVV @VVV \\...
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0answers
28 views

$R$-Orientability of a Topological Manifold

Let $M$ be a $n$-manifold and $R$ a PID ring. $M$ is then called $R$- orientable if for each $x \in M$ there exists a family of generators $[M]_x \in H_n(M, M - \{x\}) \cong H_n(\mathbb{R}^n, \mathbb{...
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2answers
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Bredon's Cone Construction

Bredon has a slick proof based on the cone construction, that if $X$ is a contractible topological space, then $H_i(X)=0$ for $i\neq 0$. My question has to do with a small detail: Let $F:X\times I\...
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2answers
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How to show $f^{-1}(D)/f^{-1}(C)\simeq D/C$ by the language of Abelian category?

Let $\mathscr C$ be an Abelian category and $f:A\to B$ an epimorphism in $\mathscr C$. Let $g:C\to D$ and $h:D\to B$ be monomorphisms in $\mathscr C$, thus $C,D$ are subobjects of $B$. Let $f^{-1}(...
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0answers
31 views

the form of classes in $K(X)$ [Atiyah]

Anyone can explain me why taking two bundles $E,F$ we can conclude that any element in $K(X)$ is of the form $E-F$? It is mentioned in Atiyah on the bottom of the page $43$ (I can put screen if needed)...
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1answer
50 views

Relation Between Homologies of $X$ and it's Suspersion $\Sigma X$

I'm reading how May's "A Concise Course in Algebraic Topology" and I have some questions about following proof: LEMMA (see pages 97-99). Here the excerpt: We have the wedge product $X= \vee_i S^n$ ...
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1answer
21 views

triangulation of a circle and the way to solve a problem

Consider the circle $S^1$ with multiplication given by the complex numbers. Prove that the map $f(x) = x ^n$ , $n$ a positive integer, has degree $n$. What is the degree of the map $g(x) = 1/x$. ...
1
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1answer
41 views

Metaplectic group is the unique double covering of symplectic group

The proof that $Mp(2n,\mathbb{R})$ is the unique connected double cover of $Sp(2n,\mathbb{R})$ uses the fact that the fundamental group of the latter is infinite cyclic (the integers). I have not ...
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0answers
50 views

The three definitions of Cech cohomology (simplical complex vs. presheaf vs. sheaf)

I came across the following three definitions of Cech cohomology group of a topological space $X$: [Source: Munkres, Elements of Algebraic Topology, pp. 437]. The Cech cohomology group of $X$ in ...
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0answers
23 views

looking for flabby sheaf resolutions

I am looking for manipulable flabby resolution of the sheaf of top-degree forms (let say on a complex manifold $X$) which is not canonical, i.e. not the Godement resolution. Does it exist any ...
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0answers
29 views

Poincaré dual of the generators of $H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)$

We know $H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)=\mathbb{Z}_4$. So there are two classes of $\mathbb{Z}_4$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$. Wha are the Poincaré dual $(5-d)$-...
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2answers
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Proof in Lee's Introduction to Topological manifolds

In his book, Lee has what appears to be a very simple proof of the fact that if $f:I\to X$ is a path in $X$, then $f^{-1}\sim -f$ where $f^{-1}(t)=f(1-t).$ He takes $\Delta_2$ to be the simplex with ...
2
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0answers
42 views

Poincaré dual of the trivial class $H^1(\mathbb{RP}^5,\mathbb{Z}_2)$

Let $$a \in H^1(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2,$$ When $a' \in H^1(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2$ is a nontrivial generator, the Poincaré dual (4-manifold generator) PD$(a')$ of ...
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0answers
38 views

Motivation for cochains, cocycles and coboundaries

I am beginning to learn cohomology, namely in the small and functional appendix of Silverman's book, The Arithmetic of Elliptic Curves. While the theory is efficiently presented and I have no problem ...
0
votes
1answer
43 views

How to show a morphism $f^•: A^•\to B^•$ of complexes induce the morphism of cohomology objects $H^i(A^•)\to H^i(B^•)$?

Let $\mathscr C$ be a Abelian category, how to show a morphism $\varphi^•: A^•\to B^•$ of complexes induce the morphism of cohomology objects $H^i(A^•)\to H^i(B^•)$? $A^•:\qquad \cdots\to A^{i-1}\...
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0answers
27 views

de Rham cohomology of vector bundles: advantages of different definitions of compact vertical cohomology

I'm currently reviewing the Thom isomorphism and the de Rham cohomology of vector bundles over a compact manifold $M$. I'm familiar with the case of topological disk bundles, where the Thom class ...
3
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0answers
41 views

Poincaré dual of $H^1(M,\mathbb{Z}_2)$ for a $\frac{\mathbb{CP}^2\times S^1}{\tau}$

Given a $M=\frac{\mathbb{CP}^2\times S^1}{\tau}$, where $τ$ acts as $−1$ on the sphere $S^1$ and a complex conjugation on complex projective space $\mathbb{CP}^2$. See Dold, Albrecht (1956), "...
3
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0answers
59 views

What is the induced isomorphism of a handle decomposition $X_k=X_{k-1}\cup H^{\lambda_k}$, $X_0=D^m$ given by the long exact Mayer-Vietoris sequence?

Let $X_k=X_{k-1}\cup_{\theta} H^{\lambda_k}$, for $k\ge 1$, be the $\lambda_k$-handle attached to a $\dim X_{k-1}$-manifold along the embedding map $\theta:S^{\lambda_k-1}\times D^{\dim X_{k-1}-\...
2
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0answers
20 views

Action of quadratic Casimir on cochain/cohomology spaces

Let $\mathfrak g$ be a semisimple Lie algebra and let $M$ be a nontrivial simple $\mathfrak g$-module. Then quadratic Casimir of $\mathfrak g$ acts on $M$ as multiplication by a nonzero scalar $c_M$. ...
5
votes
1answer
44 views

How are the homology groups of map space related to the “factors”?

If $X$ and $Y$ are reasonably nice spaces (CW-complexes, simplicial complexes, manifolds...), is there a relation between the (co)homology of the space of the continuous maps $\mathcal{F}(X, Y)$ (...
1
vote
1answer
71 views

Submanifold generator and its Poincaré dual in a 5-dimensional real projective space

1) What is the nontrivial 1-dimensional sub-manifold generator $M^1$ in $\mathbb{RP}^5$ of $H^1(\mathbb{RP}^5,\mathbb{Z}_2)$? How to visualize it where this $M^1$ sits in $\mathbb{RP}^5$? 2) What ...
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0answers
19 views

Is there an established definition of the simple complex associated to a triple (or higher) complex?

The simple complex associated with a commutative double complex has a well-established definition. The $i$th term is the sum of terms in the couple complex with indices summing to $i$, and the maps ...
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0answers
10 views

Virtual finite cohomology implies finite cohomology

Let $G$ be a torsion-free pro-$p$ group, and let $H$ be an open subgroup of $G$. Suppose both $H$ and $G$ have finite cohomological dimension. What I want to show is: If all groups $H^i(H, \mathbb{...