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Abh
Abh
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$\newcommand{\ab}{\mathrm{ab}}$Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, filling in holes) to arrive at a space $X^{\ab}$ whose fundamental group is the abelianization of $X$: $\pi_1(X^{\ab}) = \pi_1(X)^{\ab}$.

What is this procedure called? What are the operations involved? Can this always be done in a systematic manner?

I'd like to know how many homotopically distinct $X^{\ab}$ we'd get in this way, expressed in terms of some measure of complexity of $\pi_1(X)$.

Also, a basic question: how are the cohomology groups of $X$ and $X^{\ab}$ related?

Ultimately, I'd like a description of the topological simplification along all the steps of the derived series of $\pi_1(X)$.

$\newcommand{\ab}{\mathrm{ab}}$Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, filling in holes) to arrive at a space $X^{\ab}$ whose fundamental group is the abelianization of $X$: $\pi_1(X^{\ab}) = \pi_1(X)^{\ab}$.

What is this procedure called? What are the operations involved? Can this always be done in a systematic manner?

I'd like to know how many homotopically distinct $X^{\ab}$ we'd get in this way, expressed in terms of some measure of complexity of $\pi_1(X)$.

Also, a basic question: how are the cohomology groups of $X$ and $X^{\ab}$ related?

Ultimately, I'd like a description of the topological simplification along all the steps of the derived series of $\pi_1(X)$.

$\newcommand{\ab}{\mathrm{ab}}$Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, filling in holes) to arrive at a space $X^{\ab}$ whose fundamental group is the abelianization of $X$: $\pi_1(X^{\ab}) = \pi_1(X)^{\ab}$.

What is this procedure called? What are the operations involved? Can this always be done in a systematic manner?

I'd like to know how many homotopically distinct $X^{\ab}$ we'd get in this way, expressed in terms of some measure of complexity of $\pi_1(X)$.

Ultimately, I'd like a description of the topological simplification along all the steps of the derived series of $\pi_1(X)$.

formatting, changed title (it looked like asking about a topology on the group itself)
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YCor
YCor
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Topology of the Topological operations corresponding to abelianization of the fundamental group

Suppose$\newcommand{\ab}{\mathrm{ab}}$Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, filling in holes) to arrive at a space $X^{ab}$$X^{\ab}$ whose fundamental group is the abelianization of $X$: $\pi_1(X^{ab}) = \pi_1(X)^{ab}$$\pi_1(X^{\ab}) = \pi_1(X)^{\ab}$.

What is this procedure called? What are the operations involved? Can this always be done in a systematic manner?

I'd like to know how many homotopically distinct $X^{ab}$$X^{\ab}$ we'd get in this way, expressed in terms of some measure of complexity of $\pi_1(X)$.

Also, a basic question: how are the cohomology groups of $X$ and $X^{ab}$$X^{\ab}$ related?

Ultimately, I'd like a description of the topological simplification along all the steps of the derived series of $\pi_1(X)$.

Topology of the abelianization of the fundamental group

Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, filling in holes) to arrive at a space $X^{ab}$ whose fundamental group is the abelianization of $X$: $\pi_1(X^{ab}) = \pi_1(X)^{ab}$.

What is this procedure called? What are the operations involved? Can this always be done in a systematic manner?

I'd like to know how many homotopically distinct $X^{ab}$ we'd get in this way, expressed in terms of some measure of complexity of $\pi_1(X)$.

Also, a basic question: how are the cohomology groups of $X$ and $X^{ab}$ related?

Ultimately, I'd like a description of the topological simplification along all the steps of the derived series of $\pi_1(X)$.

Topological operations corresponding to abelianization of the fundamental group

$\newcommand{\ab}{\mathrm{ab}}$Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, filling in holes) to arrive at a space $X^{\ab}$ whose fundamental group is the abelianization of $X$: $\pi_1(X^{\ab}) = \pi_1(X)^{\ab}$.

What is this procedure called? What are the operations involved? Can this always be done in a systematic manner?

I'd like to know how many homotopically distinct $X^{\ab}$ we'd get in this way, expressed in terms of some measure of complexity of $\pi_1(X)$.

Also, a basic question: how are the cohomology groups of $X$ and $X^{\ab}$ related?

Ultimately, I'd like a description of the topological simplification along all the steps of the derived series of $\pi_1(X)$.

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Abh
Abh
  • 79
  • 2

Topology of the abelianization of the fundamental group

Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, filling in holes) to arrive at a space $X^{ab}$ whose fundamental group is the abelianization of $X$: $\pi_1(X^{ab}) = \pi_1(X)^{ab}$.

What is this procedure called? What are the operations involved? Can this always be done in a systematic manner?

I'd like to know how many homotopically distinct $X^{ab}$ we'd get in this way, expressed in terms of some measure of complexity of $\pi_1(X)$.

Also, a basic question: how are the cohomology groups of $X$ and $X^{ab}$ related?

Ultimately, I'd like a description of the topological simplification along all the steps of the derived series of $\pi_1(X)$.