Tagged Questions

Use this tag if your question involves some type of homology, including (but not limited to) simplicial homology, singular homology, or group homology.

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What is $C_n(X)$?

This article says the following: Let $X$ be a triangulated space and let $C_n(X)$ be a real vector space with $n$-simplices $[x_0,x_1,x_2,\dots,x_n]$. Each different combination of $x_i's$ forms a ...
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Applying the functor $H_*$ to the inclusion sequence $A\rightarrow B\rightarrow C$

Does applying the functor $H_*$ to the sequence of inclusions $A\rightarrow B\rightarrow C$ induce a map $\phi_3: H_*(B)\rightarrow H_*(C )$, such that if $\phi_1:H_*(A)\rightarrow H_*(B)$, and ...
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turning a map into a fibration

In Allen Hatcher's book Spectral Sequence page 29 Example 1.18, What means "turning the map into a fibration" and convert a map into a fibration"? Given a map $f:X\to Y$, $f$ is not necessarily a ...
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Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
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homology and cohomology with coefficients of ring and field [closed]

(1). Let $R$ be a ring. Let $X$ be a topological space. Then $H^n(X;R)$ is a module over $R$. Also $H_n(X;R)$ is a module over $R$.Is this statement correct? (2). Let $F$ be a field. Let $X$ be a ...
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Significance of homology groups of a topological space

I am studying homology groups of topological spaces. In books I have found that the $n$th homology group counts the number of "$n$-dimensional holes" which exist in that space. If I consider homology ...
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What is explicit form of this kernel?

Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $F$ and $S$ be a free group such that $F/R=G$ and $S/R=N$ for some normal subgroup $R$ of $F$. The map from $N \rtimes G$ to $G$ given by ...
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Why *formal* linear combination?

In Lee's Introduction to topological manifolds on page 340 he writes that an element of $C_p(X)$ can be written as a formal linear combination of singular $p$-simplices. Similarly, on Wikipedia's ...