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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Hatcher exercise 43c, section 2.2: If $X$ is a CW complex with finitely many cells in each dimension, then $H_n(X;G)$ is a direct sum...

Below is the question. I will briefly describe how I solved parts a and b, then say why I'm stuck on c. There are other questions on this site which address this exercise, but none dealing with part c ...
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Proof of $\tilde{H}_*(X) \cong H_*(X, x_0)$.

Let $X$ be a topological space and $x_0 \in X$. I want to show that the relative homology group, $H_*(X, x_0)$, is isomorphic to the reduced homology group, $\tilde{H}_*(X)$. By considering the long ...
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Explicit description of the homology of the n-sphere

I'm currently studying the homology of manifolds (so it's mainly orientation and duality questions) and I was wondering how to explicitly compute homology of spaces and particularly $n$-spheres. ...
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Conditions for null-homotopic chain map in a specific example

I am trying to calculate the ring $Ext^{\bullet}_R(k,k)$ where $R=k[x,y]/(xy)$ and $k$ is regarded as an $R$-module via $x$ and $y$ acting as zero. I thought I was done but then to my demise I found a ...
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Computation of $\tilde{H}_k(\bigvee_{j = 1}^N S^n)$.

In my algebraic topology course, we state the following proposition $$\tilde{H}_k(\bigvee_{j = 1}^N S^n) \cong \begin{cases} \mathbb Z^N & \text{if } k = n,\\ 0 & \text{else}, \end{cases}$$ ...
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Proof of $\tilde{H}_k(X \lor Y) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y)$.

In my algebraic topology course, we showed the following statement: Let $X, Y$ two topological spaces, $(x_0, y_0) \in X \times Y$ and $X\lor Y$ the wedge product of $(X, x_0)$ and $(Y, y_0)$. If $x_0$...
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To understand whether a module is flat, injective or projective through a explicit example

Suppose a ring $R = \mathbb{C}[x, y]$, An ideal $I = (y^2 − x^3 + x^7)$ and $M = (R/I )_I \oplus (R/I )$. We can easily check that $R/I$ is an integral domain thus $I$ is a Prime Ideal. Hence ...
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A question about Mayer-Vietoris sequence [duplicate]

Let $A$ and $B$ be two open sets in $\mathbb{R}^n$ ($n>1$), with $A \cup B=\mathbb{R}^n$ and $A\cap B\not=\emptyset$. There is a version of the Mayer-Vietoris sequence for reduced homology that ...
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Torsion in the integral (co)homology of Eilenberg-MacLane spaces

I've had trouble finding a reference. I know that the classifying space of a group $G$ is an example of an Eilenberg-MacLane space $K(G,1)$, so that the cohomology of $K(G,1)$ is group cohomology. If ...
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Sheaf homology defined in terms of Tor

By the general philosophy of cohomology, cohomology is essentially derived $\operatorname{Hom}$ (i.e. $\operatorname{Ext}$), and homology should be derived tensor product (i.e. $\operatorname{Tor}$). ...
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Singular homology with coefficients in a ring versus in an abelian group

As described here and here the singular homology of a topological space $X$ with coefficients in a ring $R$ is given by a bunch of $R$-modules $H_n(X,R)$. However, sometimes I see people talking about ...
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Equivalence of homology theories via acyclic models

I'm looking for a proof of "any two homology theories are equivalent" (obviously with some other hypothesis) via the Acyclic Models Theorem. I know that this is an application of the Acyclic ...
I've been stuck on one step of a proof, and am hoping someone will have a helpful hint/explanation :) Here is the setting:we have two Lie groups $K \hookrightarrow T$ acting on a manifold $M$. We can ...
The singular homology of a space $X$ is defined to be the homology of the chain complex {\displaystyle \ldots {\stackrel {}{\longrightarrow }}\mathbb Z[Sing_2(X)]{\stackrel {}{\longrightarrow }}\...