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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How do I know when a form represents an integral cohomology class?

Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class. I would like to understand the meaning of the following sentence…
student
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The First Homology Group is the Abelianization of the Fundamental Group.

I am trying to understand the proof of the following fact from Hatcher's Algebraic Topology, section 2.A. Theorem. Let $X$ be a path connected space. Then the abelianization of $\pi_1(X, x_0)$ is isomorphic to $H_1(X)$. I am having trouble…
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Why is there "no analogue of $2i\pi$ in $\mathbf C_p$"?

In his paper Fonctions L p-adiques, Pierre Colmez says: Tate a montré qu'il n'existait pas dans $\mathbf C_p$ d'analogue $p$-adique de $2i \pi$ et donc par conséquent que les périodes $p$-adiques des variétés algébriques ne pouvaient pas vivre dans…
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Does taking the direct limit of chain complexes commute with taking homology?

Suppose I have a directed system $C_i$, $i\in\mathbb{N}$ of chain complexes over free abelian groups (bounded below degree $0$) $$C_i=0\rightarrow C^{0}_{(i)}\rightarrow C^{1}_{(i)}\rightarrow\cdots\rightarrow C^{n-1}_{(i)}\rightarrow…
Dan Rust
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Topological vs. Algebraic $K$-Theory

Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras associated with $X$? What is the algebraic analogue of…
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Integral classes in de Rham cohomology

If $M$ is a differentiable manifold, De Rham's theorem gives for each positive integer $k$ an isomorphism $Rh^k : H^k_{DR}(M,\mathbb R) \to H^k_{singular}(M,\mathbb R)$. On the other hand, we have a canonical map $H^k_{sing}(M,\mathbb Z) \to…
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What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?

Let $X$ be a non-singular projective variety over $\mathbb{Q}$. Consider on the one hand $H^i_B(X(\mathbb{C}),\mathbb{Z}_\ell)$ the singular cohomology with value in $\mathbb{Z}_\ell$, and on the other hand $\varprojlim…
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Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose $X,Y$ are two connected CW complexes and $f:X\to Y$ is a…
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In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

I started reading about $p$-adic Hodge theory in the notes of Brinon and Conrad. I quote (page 7): The goal of p-adic Hodge theory is to identify and study various “good” classes of $p$-adic representations of $G_K$ for p-adic fields $K$,…
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Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have the following isomorphism for each $k \leq m$ …
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Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you read well the answer, it starts by stating that it…
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Why group cohomology and not group homology?

In algebraic topology, one studies the homology and cohomology of spaces. However, when we study group homology/cohomology, we almost exclusively talk about cohomology. Why is this? Is there an aesthetic reason to prefer cohomology in this…
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The "need" for cohomology theories

In many surveys or introductions, one can see sentences such as "there was a need for this type of cohomology" or "X succeeded in inventing the cohomology of...". My question is: why is there a need to develop cohomology theories ? What does it…
MarcSimon
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Why Cohomology Groups?

Why do we need cohomology groups? Homology groups are easier to compute and given two topological spaces, there is an isomorphism in homology groups if and only if there is an isomorphism in cohomology groups. So why do I need them?
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Toy sheaf cohomology computation

I asked this question a while back on MO : One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$) I am curious, as a sort of followup if anyone can suggest: A reference where small…