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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3
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1answer
29 views

Product of inexact differential forms is inexact

Suppose we have a product manifold $M = M_1 \times M_2$. Let $\omega$ be a closed but inexact form on $M_1$ and $\eta$ a closed but inexact form on $M_2$. Then the claim is that $$\omega \wedge \eta$$ ...
4
votes
1answer
46 views

Property similar to connectedness

Recall that $X$ is connected if $X$ cannot be written as the union of nonempty open sets with empty intersection. Consider the following similar property: $X$ is good if $X$ cannot be written as ...
0
votes
0answers
22 views

$\mathbb{P}_{\mathbb{C}}^3$ is not isomorphic to $S^2 \times S^4$

I have been trying to solve this exercise given by my prof. The hint is to show that every $2$-form $w$ on $S^2 \times S^4$ is s.t. $w \wedge w = 0$, while this is not true in case of ...
0
votes
0answers
14 views

Compact Poincaré dual of $S^{n-1}$ in $\mathbb{R}^n \backslash \{0\}$

I have been asked to find a sub manifold $S \subset M := \mathbb{R}^n \backslash \{0\} $ s.t. its compact Poincaré dual is a basis of the first cohomology group $H^1_c(M) = \mathbb{R}$. Now, $S$ must ...
0
votes
0answers
10 views

Computing the khovanov polynomial of the trefoil knot

I am reading the following article about khovanov polynomials of bar nathan: http://arxiv.org/abs/math/0201043 So far i have succeeded computing these polynomials for the hopf link, trivial link and ...
1
vote
1answer
46 views

Filling in Proof: Well-definedness of depth(I,M).

From Eisenbud's Commutative Algebra with A View Toward Algebraic Geometry (Theorem 17.4): Let $M$ be a finitely generated $R$-module, where $R$ is Noetherian. If $$r= \min \{i : H^i(M\otimes ...
1
vote
0answers
19 views

Help understanding remark on Greenberg and Harper text on cohomology chapter

I am working through Greenberg and Harper text and I don't understand a remark. I think it should be very easy to gasp but I don't see what it means: My problem is with the remark 23.0. Could ...
1
vote
0answers
37 views

Why abelian groups instead of modules in Algebraic Topology

I am studying Algebraic Topology, homology and cohomology to be concrete. I am reading\working through Hatcher, Rotman, Harper and sometimes I combine them with other books when none of them give a ...
1
vote
0answers
36 views

Computing homology of a torus

I'm trying to calculate homology groups of a a torus using Meyer-Vietoris sequence. Let $A,B$ be a half of a torus homeomorphic to $S^{1} \times I $. Let's enlarge them so that they intersect and $A ...
2
votes
0answers
33 views

Homology, addition of homology classes in construction of Poicare Sphere

I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one exercise. I have spoken with a professor and he encouraged me to ask here or look for ...
0
votes
1answer
20 views

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for $k = n$ and $k = n-1$?

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for: (1) $k = n$, and (2) $k = n - 1$. Poincaré Duality tells us that for $M$ a closed ...
1
vote
0answers
10 views

Rational homology of $\Omega^{n+1}\Sigma^{n+1}X$

I want to know how compute, by induction and using the Serre spectral sequence for homology, $H_*(\Omega^{n+1}\Sigma^{n+1}X, \mathbb{Q})$. I know that I have to use the path-loop fibration $$ ...
2
votes
0answers
50 views

Homological description of the degree of a map to $\mathbb P^n$

Let $f \colon X \to \mathbb P^n$, $n \geq 2$, be a holomorphic map from a compact Riemann surface $X$ and whose image $f(X)$ is a smooth projective curve. There are two notions of degree for such a ...
3
votes
1answer
34 views

Fundamental Group and DeRahm Cohomology from Group of Covering Transformations

Old qual problem here, test tomorrow in topology and we barely got to DeRahm Cohomology so I'm not sure how to do this. Let $G$ be the group of transformations of $\mathbb{R}^3$ generated by ...
2
votes
0answers
9 views

When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
0
votes
0answers
22 views

Cellular homology for 3-sphere and $L^3$ lens space

When trying to follow the professor's computation of the cellular homology for the lens space $L^3(4)=S^3/\mathbb{Z}_4$ I became aware I had some trouble understanding the definition of the (cellular) ...
0
votes
0answers
34 views

Homology Poincare Homology Sphere by Mayer-Vietoris

I am working through some pages of Dale Rolfsen, Knots and Links, AMS Chelsea Publishing, 2003, in order to understand Dehn approach to the original Poincaré conjecture. To be concrete with what I ...
0
votes
1answer
19 views

Show that the maps are chain homotopic

Let $\Delta _{2}$ be a 2-simplex, $I=\left [ 0,1 \right ]$. Given are two maps $i_{0}:\Delta _{2}\rightarrow \Delta _{2}\times I$, defined by $x \mapsto (x,0)$ and $i_{1}:\Delta _{2}\rightarrow ...
0
votes
0answers
41 views

Knot theory and homology

What is the best way to learn about homology in knot theory? I am looking for a introductory book or resource, I dont know any homology, would I need to read a book about this first? If so, which?
1
vote
0answers
20 views

Why is $H_{DR}^p(M,\mathbb{C})\cong H_{DR}^p(M,\mathbb{R})\otimes_\mathbb{R}\mathbb{C}$

This question is related to my previous question. The answers to that question inspired a new question, namely For a complex manifold $M$, why is $H_{DR}^p(M,\mathbb{C})\cong ...
1
vote
0answers
24 views

De Rham cohomology ring of flag bundles/manifolds in Bott and Tu

I'm trying to understand the result for the de Rham cohomology ring of flag manifolds in Differential Forms in Algebraic Topology by Bott and Tu. I'm sort of starting from the result and working ...
1
vote
1answer
42 views

Poincaré Duality in Middle Dimension

I am reading a paper that states the following theorem without proof: Poincaré duality in middle dimension: Let $M$ be a connected oriented manifold of even dimension $2d$. Then the cup product ...
0
votes
1answer
28 views

Relevance of the coeficient ring in various cohomologies

I'm reading "Principles of algebraic geometry" by Griffiths and Harris. While reading the first chapter, I keep running into the same problem, which I'll illustrate using some examples: Proven: "On ...
0
votes
1answer
35 views

Sphere with four points deleted

What is the universal cover of the sphere with four points deleted and a non-trivial abelian fundamental group?
1
vote
1answer
41 views

Method to calculate the de Rham cohomology of $\mathbb{R}\mathrm{P}^n$

I'm trying to follow through a method to calculate the de Rham cohomology groups of $\mathbb{R}\mathrm{P}^n$ from the de Rham cohomology groups of $S^n$. I'm trying to show that differential k-forms ...
2
votes
1answer
49 views

Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
1
vote
1answer
17 views

Integral of $r$-form over singular $r$-simplex

I define a singular $r$-simplex as a smooth map $f$ from the standard $r$-simplex $\Delta^r$ into a smooth manifold $M$. The integral of an $r$-form $\omega$ over an $r$-simplex is then defined as $$ ...
2
votes
0answers
22 views

Invariant cohomology for non-compact groups

Suppose I have a compact $G$-space $M$, and a differential form $\omega$ on $M$ with the property that $$ \forall g\in G\quad g\omega = \omega + d\lambda_g, \quad(*) $$ i.e. $g\omega$ is cohomologous ...
0
votes
1answer
66 views

Real singular (co)homology of projective plane/Klein bottle without Mayer-Vietoris/Van-Kampen [closed]

I'm reading differential geometry books and trying to learn the singular (co)homology with real coefficients of the Klein bottle and projective plane by their fundamental rectangles. Here is the ...
0
votes
0answers
69 views

Which homology groups of a closed orientable 6-manifold can be isomorphic to $\mathbb{Z}^3$?

List all $i$ for which there is a closed orientable $6$-manifold $M$ with $H_i(M) =\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$ I am working on an old exam problem and this one stumped me. ...
2
votes
1answer
55 views

Question about connections on the dual bundle.

Let $E \to M$ be a vector bundle with connection $\nabla$. Extend $\nabla$ to $E^*$ and $E^* \otimes E$ in the regular fashion. Is $\text{Id} \in E^* \otimes E$ necessarily parallel?
1
vote
0answers
24 views

Determine the chain complex $(C_{*}(X),\delta )$ associated to that cell decomposition

Let $X = K \times \mathbb{R}P^3$ be the product of the Klein bottle with the 3 -dimensional projective space. 1) Find a CW-decomposition of $X$. 2) Determine the chain complex $(C_{*}(X),\delta )$ ...
3
votes
1answer
52 views

When are homology groups the trivial group?

I've noticed that all the spaces $X$ whose (singular) homology I've computed or seen computed have $H_n(X)=0$ whenever $n$ is greater than the dimension of $X$. So I have the following conjecture: ...
0
votes
0answers
12 views

Computing boundary homomorphisms in cellular chain complex of lens spaces

I am working on my own through Hatcher's book and I am having trouble while understanding the computations of cellular homology for the CW-complex structure of Lens spaces. It is on page 145. I upload ...
0
votes
1answer
35 views

Homology group of Klein bottle via Mayer Vietoris. Explanation of “ Since the boundary circle of a Mobius band wraps twice around the core circle ”

My qeustion about first homology group of Klein bottle via Mayer Vietoris sequence . i have exact sequence below $ 0 \xrightarrow{} H_1( S^1) \xrightarrow{(i,j)} H_1(M) \oplus H_1(M^\prime ...
0
votes
0answers
34 views

Understanding the cup product definition

I am trying to learn a little cohomology, and am having some trouble with this definition of the cup product that I found: $$(f\cup g) (\sigma) =f\left(\sigma_{[v_0, \ldots, v_k]} ...
1
vote
1answer
77 views

Computational complexity of solving linear diophantine equations?

Is there any good complexity upper bound for checking satisfiability of a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$? I found some estimate on computing the Smith Normal Form $N$ such that ...
1
vote
1answer
15 views

Condition on compact subsets implying null singular homology

This is a review question I'm doing for an upcoming exam. Consider $X$ a Hausdorff space, and $(H_*(X),\partial_*)$ its singular homology. I must prove that, given $[z]\in H_p(X)$ there exists a ...
1
vote
0answers
18 views

Why is are the simplicial 1-chains $[A,B] \neq -[B,A]$?

This is a really simple question that I think I have answered, but I'm not altogether satisfied and would like confirmation or an alternative. We define the simplices $$ \begin{align} [A,B] : ...
3
votes
0answers
30 views

computing $\pi_1 S^1$ from a spectral sequence

In most of my calculations that I have done based off of mosher and tangora, calculations have proceeded by knowing for example that the fiber of some fibration is say a $1-sphere$. From this we are ...
2
votes
0answers
15 views

Compact Vertical Cohomology and Euler Class of CP1

First of all, please excuse my English. I'm not native Englsih speaker, so you will see so many grammer mistakes, I just only hope that my mistkae wouldn't effect what I want to mean. Hi, recently ...
0
votes
0answers
12 views

Ratio of dimension sizes in a pure $2$-dimensional simplicial complex

I'm listening to Alex Lubotzky's YouTube lecture on Cohomology and Computer Science, and during a proof he makes the claim: $\displaystyle\frac{|X(2)|}{|X(1)|} \sim \frac{n}{3}$, where $n$ is the ...
1
vote
1answer
28 views

The value of the integral of the curvature of a complex line bundle

I am trying to show that if $L \to M$ is a complex line bundle endowed with a connection $\nabla$, $F$ is the curvature form, and $S \in M$ a closed surface, then $\int \limits _S F \in 2 \pi \textrm ...
0
votes
0answers
16 views

Cohomology of Grassmanian: pairing with fundamental class

Let $Gr(k, V)$ be a Grassmannian with $\dim V=n$, and $S$ be a tautological bundle over $Gr(k, V)$, so $\operatorname{rank} S=k$. Then the cohomology ring $H^*(Gr(k, V))$ is generated by Chern classes ...
2
votes
2answers
51 views

Representable homology classes on smooth manifolds

Let $X$ be a closed (compact without boundary) smooth manifold. We can consider its singular homology $H_*(X,\mathbb{Z})$. Let $H_{k}(X,\mathbb{Z})$ be the $k$-th singular homology group of $X$ and ...
0
votes
0answers
27 views

Generalization of a usual complex analysis fact

Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
2
votes
1answer
35 views

Representing the $2$-homology classes of a $4$ manifold. Last passage of a Proof

I found a few occurrences of the same proof about representability of elements of $H_2(M,\mathbb{Z})$ for $M$ a closed orientable smooth $4$-manifold. All of them stop at the very end claiming that ...
3
votes
0answers
14 views

Degree of quaternion product composed with two maps of $S^3$

Let $S^3$ denote the unit quaternions with multiplication $\mu:S^3\times S^3\rightarrow S^3$.Show that if $f_1,f_2:S^3\rightarrow S^3$ are given maps,that the composition ...
4
votes
0answers
26 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces ...
11
votes
2answers
157 views

Why are we interested in cohomology?

I've been studying algebraic topology for over half a year now and came across alot of different topics of it (fundamental groups, Van Kampen, singular homology, homology theory, Mayer Vietoris, ...