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.

learn more… | top users | synonyms (2)

1
vote
0answers
13 views

Appropriate Generalization of Statement about Pure Subgroups to Pure Submodules

I have been working in a book on Homology by Hilton & Stammbach, wherein they introduce the idea of a "pure sequence of Abelian groups", which is a short exact sequence of Abelian groups ...
2
votes
1answer
39 views

All cohomology of quadrics comes from algebraic cycles

I've read in a number of place now the statement that all cohomology of quadrics (complex projective ones) comes from algebraic cycles, but I cannot find any source for this. So I hope someone here ...
1
vote
1answer
32 views

Homology homomorphism induced by linear isomorphism

Let $F: R^n \rightarrow R^n $ be a linear isomorphism. I'd like to prove that induced homomorphism $H_n(R^n, R^n \setminus \left\{ 0 \right\}) \rightarrow H_n(R^n, R^n \setminus \left\{ 0 \right\})$ ...
1
vote
1answer
57 views

If $i$ is an inclusion why is the induced $i_*$ an epimorphism

Given the following exact homology sequence of a pair. This is in Example 2 (page 134) from Munkres. This is where I am always stuck computing homology using exact sequence. I cannot grab the last ...
1
vote
1answer
29 views

Good introduction to cohomology of spaces?

I'm trying to read chapter $3$ of Hatcher but I find it a bit difficult to read. I really only made it through the first two chapters because I had in-class lectures to go along with the reading. Does ...
0
votes
0answers
30 views

Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
0
votes
1answer
44 views

Show $\mathbb{C}P^n$ is a $2n-$manifold [in singular homology theory]

There is a Theorem in the book that says: The space $\mathbb{C}P^n$ is CW complex of dimension $2n$. I wonder some questions: Is there any Theorem or result that if a space has CW complex ...
1
vote
1answer
23 views

Induced homomorphism of a covering space

How can I determine what's the induced homology homomorphism of a covering $S^{n} \rightarrow RP^{n}$? I suppose that a Hurewicz homomorpism would be pretty effective, but since I know nothing about ...
2
votes
0answers
14 views

Intuition Homology and homology groups

Could someone give an intuition on the concepts of homology and homology groups? I have been reading the definition of these, but don't have a clear understanding of them. Thanks!
4
votes
0answers
48 views

Cohomology of local systems on spaces with the same homotopy type

Suppose we have a homotopy equivalence $f: X \to Y$ with homotopy inverse $g: Y \to X$, and a local system (i.e. a locally constant sheaf) $\mathcal{V}$ on $Y$. In this situation, are any of the ...
0
votes
0answers
44 views

Dimension of the zero-th cohomology for an ample line bundle [on hold]

Given an ample line bundle L on an abelian variety X, we have that $\dim H^0(X,L)>0$. Why?
0
votes
0answers
15 views

Weibel Exercise 4.5.6

Let $R$ be a regular local ring with residue field $k$. Show that $Tor_{p}^{R}(k,k) \cong Ext_{R}^{p}(k,k) \cong \Lambda^{p}k^n \cong k^{{n}\choose{p}}$. Because of Koszul Resolution and we know ...
1
vote
2answers
44 views

Cohomology of a mapping torus

How does the monodromy in a mapping torus $K_{\phi}$ affect the de Rham cohomology, if at all? Maybe this is naive, but I don't see how twisting one of the ends of $K\times I$ via the diffeomorphism ...
0
votes
1answer
36 views

Computing homology group using Mayer-Vietoris sequence

Suppose I am given an exact sequence: $$0\to G\xrightarrow{f} \mathbb{Z} \xrightarrow{g} \mathbb{Z} \xrightarrow{h} H\to 0 $$ where the first $\mathbb{Z}=H_3(A\cup B)$ and the second ...
1
vote
1answer
34 views

Low torsion in orientable manifolds?

The final sentence on page 170 of Stillwell's Classical Topology an Combinatorial Group Theory is: Poincaré justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only ...
2
votes
1answer
51 views

Nonzero-homologous simple loop in Mobius band only winds once

I have a question as follows: Let $C$ be a closed curve in the Mobius band without self intersections. Prove that if $C$ is of non-zero homology, i.e., $C$ does not bound any face, then $C$ winds only ...
1
vote
0answers
17 views

Reference request: a Künneth spectral sequence map from equivariant K-theory to cohomology

The analogue Künneth formula for $G$-equivariant cohomology can be obtained as the Eilenberg–Moore spectral sequence of a pullback $\require{AMScd}$ \begin{CD} (X \times Y)_G @>>> ...
5
votes
0answers
67 views

Barycentric subdivision proof

In the proof of barycentric subdivision in singular homology, we take the subdivision operator $b: C_q(X) \to C_q(X)$, and do some algebra to define an operator $b^\infty$, which applies $b$ "as many ...
1
vote
0answers
35 views

Prove that the induced map is of degree n

Let $P(z)$ be a complex polynomial of degree n. $$P:S^2 \rightarrow S^2 $$ $S^2 - p_0 \cong C$ (stereographic projection) and $P(\infty)=\infty$. I'd like to prove that $P_{*}:H_2(S^2) \rightarrow ...
7
votes
1answer
135 views

Detail in the proof that sheaf cohomology = singular cohomology

Theorem: If $X$ is locally contractible, then the singular cohomology $H^k(X,\mathbb{Z})$ is isomorphic to the sheaf cohomology $H^k(X, \underline{\mathbb{Z}})$ of the locally constant sheaf of ...
1
vote
1answer
54 views

Proving equality of homology of a product

I'd like to prove the following equality: $$H_i(X \times S^{n}) = H_i(X) \times H_{i-n}(X) $$ For $n=0$ it's pretty obvious, hence I'll use induction.Dividing $S^{n}$ into two hemipsheres ...
0
votes
1answer
29 views

Induced homomorphism from homology group of circle to homology group of $\mathbb{R^2-}0$ is trivial

Let $C_r$ be a circle of radius $r$ in complex plane, and let $f:C_r\to\mathbb{R^2}-0$ defined by $f(z)=z^n+a_{n-1}z^{n-1}+...+a_0$ and suppose that it has no zero on and inside the circle $C_r$. ...
2
votes
0answers
34 views

Exercise 1.5.7 in Weibel's book about mapping cone and mapping cylinder

Given a short exact sequence of chain complexes $$0\rightarrow B \xrightarrow{\ f\ }C \xrightarrow{\ g\ }D\rightarrow 0$$ The problem asks to show that there is a quasi-isomorphism $B[-1]\rightarrow ...
2
votes
3answers
62 views

Showing de Rham cohomology $H^1(S^n)$ is zero

I'm trying to find an elementary way to see that the 1st de Rham cohomology of the n-sphere is zero for $n>1$, $H^1(S^n) = 0$. This is part of an attempt to find the de Rham cohomology of the n ...
0
votes
1answer
24 views

Is this a right calculation of Homology groups?

Let $X$ be a circle $S^{1}$ and f a map $f:S^{1} \rightarrow S^{1} $; $f(z)=z^5$. I'd like to calculate homology groups of mapping torus of this space.$$$$ $[X \times I]/\cong$ can be depicted as a ...
1
vote
0answers
40 views

Homology groups of $S^1 \times (S^1 \vee S^1)$ [duplicate]

How to compute all homology groups of $S^1 \times (S^1 \vee S^1)$? Thank you. I don't see how to define a simplicial structure.
1
vote
1answer
26 views

Induced map on zeroth homology is zero

I am working through some examples on the homology of mapping torii in Hatcher's Algebraic Geometry. One thing that is confusing me is the following: I don't see why the map on the zeroth homology ...
1
vote
1answer
20 views

Excercise 1(b) on Zero-dimensional Homology in Munkres

If $\phi:C_0(K)\to \mathbb{Z}$ is an epimorphism such that $\phi\circ \partial_1=0$ then show that $$H_0(K)\cong \frac{ker\phi}{im\partial_1}\oplus\mathbb{Z}.$$ My working is since $C_0(K)$ is ...
0
votes
0answers
31 views

Homology of $X/\{x\sim f(x)\}$ where $f\colon X\to X$

Let $X$ be a space and $f\colon X\to X$ a continuous map. What tools do we have to compute the homology $H_n(X/\sim)$ where $\sim$ is defined by $x\sim f(x)$? Relative homology was my first though, ...
1
vote
1answer
30 views

Isomorphism on top cohomology implies isomorphism on homology

Let $F$ be a finite field (for example I could take $\mathbb{Z}_2$) and $f:X\longrightarrow Y$ a continuous map between compact, orientable and connected manifolds of dimension $n$. Suppose I have an ...
1
vote
0answers
25 views

w.r.t. which chain complex is $H^k_{sign}(M;R)$ computed?

This question is inspired by my previous question. People often write $H^k_{sign}(M;R)$ for the $k$th singular cohomology group with coefficients in $R$. However, I don't understand what this means. ...
0
votes
1answer
30 views

Constructing a map $H^{k}(M,\mathbb{Z})\to H^{k}(M,\mathbb{C})$

I read that on a compact oriented manifold, there is a map $$H^{k}(M,\mathbb{Z})\to H^{k}(M,\mathbb{C}).$$ I want to be sure that I have the right map in mind. We don't have an inclusion, since ...
1
vote
0answers
25 views

Homology and triangulation of open surfaces

For example I have an open disk, or an open annulus. How do I triangulate open surfaces to find their (simplicial) Homology? Well, I know that open disk and closed disk are both homotopic to a ...
2
votes
1answer
28 views

Some basic question on pasting map from a square to a Klein bottle and homology

Consider a square $S$ which edges identified as follows Let $K$ be a Klein bottle and $p:S\to K$ be pasting map. Let $X$ be the image of the interior of $S$ under $p$ and let $Y$ be the image of a ...
0
votes
0answers
35 views

Are complex subvarieties cycles in the sense of singular homology?

Given a $p$-codimensional complex subvariety $Z\subset M$ of a non singular complex projective variety $M$ of dimension $n$ we can define an element $$\int_\hat{Z}i^*\in ...
1
vote
1answer
49 views

Homology of $Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$

I want to compute the homology of $M=Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$. I think I have the answer, but I'm not sure how to make it precise. My approach is to consider the affine cover ...
1
vote
0answers
34 views

Relative Homology (Question about Example 2 in Munkres)

I have no problems for $p=0$ case and for $p\geq 2$ it is quite obvious since $C_p(K,v)=C_p(K),\forall p\geq 2$. Now the tricky is for $p=1$. Since the elements of the kernel now not anly map to $0$ ...
1
vote
1answer
24 views

Understanding proof of Universal coefficient theorem for cohomology

I am working through Cohomology chapter on Hatcher's book and I am having trouble with the proof of Universal Coefficient theorem for Cohomology. To be concrete I don't understand the last part of the ...
0
votes
2answers
46 views

Isomorphism in fundamental group implies isomorphism is homology

Let $X$ be a connected space and $f:X\longrightarrow X$ a map. Suppose $\pi_1(X)$ is an abelian group and that $\pi_1(g):\pi_1(X)\longrightarrow\pi_1(X)$ is an isomorphism. I know we can deduce that ...
1
vote
1answer
64 views

Odd map implies odd degree with homology and cohomology

Suppose $n$ is odd. Let $f:S^n\longrightarrow S^n$ be an odd function. Then it induces a map $g:P^n\longrightarrow P^n$ such that the diagram $$ \begin{array}[c]{ccc} ...
2
votes
1answer
38 views

Isomorphism on Cohomology implies isomorphism on homology

Say I am given a chain map $f:C \to D$ of complexes of (free if necessary) abelian groups. Assume that this map induces isomorphisms of cohomology with all coefficient rings. How do you prove that ...
1
vote
0answers
36 views

Questions about complexes and homology

I just learn about the simplicial and delta complexes and computing homology group. But I have a few questions: Is there any topological space which cannot be given a delta compplex structure? Is ...
0
votes
2answers
64 views

Homology of the $n$-torus using the Künneth Formula

I'm trying to apply the Künneth Formula $$H_{n}(X \times Y) \simeq \displaystyle \bigoplus_{r+s=n} H_{r}(X) \otimes H_{s}(Y)$$ to compute the homology groups of the $n$-torus. For the double torus, ...
1
vote
0answers
35 views

Weibel IHA exercise 1.2.6 : Example of total complex

1.2.6 is below; Give examples of (1) a second quadrant double complex C with exact columns such that $Tot^{\prod}(C) $ is acyclic but $Tot^{\oplus}(C)$ is not; (2) a second quadrant double complex ...
2
votes
1answer
23 views

Given that $H^1(X)=0$ on a connected space, show that all maps to $X\to S^1$ are null homotopic

Let $X$ be a path-connected, locally path-connected topological space, with $H^1(X)=0$. I would like to show that any map $f:X\to S^1$ is null homotopic, but I haven't really made any progress. ...
0
votes
1answer
67 views

Homology group of non orientable manifold

I think that if $M$ is a non-orientable, connected, compact, n-manifold, then $H_n(M,\mathbb{Z}/k)=0$ if $k\neq 2$. My proof is the following: $H_n(M,\mathbb{Z}/k)\neq 0$ then $H_n(M,\mathbb{Z}/k)$ ...
1
vote
1answer
19 views

induced homomorphism in homology

I've read a lot about induced homomorphisms in homology, but I need to see it on some examples. Let's say we have an inclusion $i: S^0 \to D^1$ and it induces the homomorphism $i_*: H_0(S^0) \to ...
1
vote
1answer
18 views

image of boundary operator in homology

I'm starting to learn homologies and I'm facing a lot of problems. We have a matrix $A$, let's say: $$ \begin{pmatrix} 1 & 2\\ 3 & 4\\ \end{pmatrix}$$ And there is a ...
3
votes
1answer
42 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 $$\omega \wedge \eta$$ ...
5
votes
1answer
86 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 ...