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.

A chain complex $(A_{\bullet}, d_{\bullet})$ is a sequence $(A_n)_{-\infty}^{\infty}$ of abelian groups (or modules) and group (module) homomorphisms $d_n : A_n \to A_{n-1}$ such that $d_{n-1}\circ d_n = 0$. This data can be represented as follows:

$$\cdots \xrightarrow{d_{n+1}} A_n \xrightarrow{d_n} A_{n-1} \xrightarrow{d_{n-1}} \cdots$$

The homology of a chain complex is the sequence of abelian groups

$$H_n = \frac{\ker d_n}{\operatorname{im}d_{n+1}}.$$

Dually, a cochain complex is a sequence $(A_{\bullet}, d_{\bullet})$ of abelian groups where $d_n : A_n \to A_{n+1}$.

There are many common types of (co)homology including simplicial (co)homology, singular (co)homology, and group (co)homology. A more extensive list can be found here.

Simplicial homology and singular homology are examples of homology theories attached to a topological space. The Eilenberg-Steenrod axioms are a collection of properties that such homology theories share.

For the more abstract aspects of (co)homology theory, the tag may be more appropriate.

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Existence of topological space which has no "square-root" but whose "cube" has a "square-root"

Does there exist a topological space $X$ such that $X \ncong Y\times Y$ for every topological space $Y$ but $$X\times X \times X \cong Z\times Z$$ for some topological space $Z$ ? Here $\cong$ means homeomorphic.
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Why would I define Alexander–Spanier cohomology?

I think I can motivate the definitions of simplicial, singular, de Rham, Čech, and sheaf (co)homology, more or less. I might want to understand bordism, and start by trying to understand submanifolds, then realize that this is really hard to do…
jdc
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how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without Mayer-Vietoris,just by Calculus. I have tried and failed.Is…
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Why homology with coefficients?

I am currently studying a bit of homology theory (on topological spaces). Let $H_n(X)$ denote the singular homology groups of the topological space $X$, then as you know we can define the singular homology with coefficients in the abelian group $G$…
Daniel Robert-Nicoud
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Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the first place), there are many more fields in…
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Cohomology of projective plane

How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
user5644
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Poincare Duality Reference

In Hatcher's "Algebraic Topology" in the Poincaré Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's easy to see that the first structure gives the…
J126
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Motivating Cohomology

Question: Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology? Why do I care: For a number of math kids I know,…
user2959
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Vanishing differential forms in cohomology

Let $X$ be a smooth differentiable manifold. Consider on $X$ a closed $p$-form $\eta$ and a closed $q$-form $\omega$, which have associated cohomology classes $[\eta] \in H^p(X)$ and $[\omega] \in H^q(X)$. Now assume their wedge product is zero in…
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Meaning of "efface" in "effaceable functor" and "injective effacement"

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means to erase it, but I'm not sure if there's another…
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Is homology with coefficients in a field isomorphic to cohomology?

is it true that when we compute homologies and cohomologies with coefficients in a field then homology and cohomology groups are isomorphic to each other? That is valid when homology groups are free with integer coefficients.
Lehi
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References for calculating cohomology rings

I am struggling to calculate homology rings. Even for a simple space such as the sphere, it is easy to calculate the cohomology, but I find it much harder to find the ring structure. (This link gives the answer for the 2-sphere, and the…
Juan S
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Why are period integrals naïve periods?

Apologies for the long question. I recall the definition of a (naïve) period according to Kontsevitch and Zagier [KS]: A (naïve) period is a complex number whose real and imaginary parts are absolutely convergent integrals of rational functions…
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Calculating the cohomology with compact support of the open Möbius strip

I am having problems calculating the cohomology with compact support of the open Möbius strip (without the bounding edge). I am using the Mayer Vietoris sequence: U and V are two open subsets diffeomorphic to $\mathbb{R}^2$ and $U\cap V$ is…
tigu
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The simplest nontrivial (unstable) integral cohomology operation

By an integral cohomology operation I mean a natural transformation $H^i(X, \mathbb{Z}) \times H^j(X, \mathbb{Z}) \times ... \to H^k(X, \mathbb{Z})$, where we restrict $X$ to some nice category of topological spaces such that integral cohomology…
Qiaochu Yuan
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