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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 $\pi_1(X)$.

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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    $\begingroup$ It's not exactly abelianization, but you might be interested in Quillen's +-construction. You can of course always add cells to $X$ to kill commutators in $\pi_1$, but I don't know if you'd like something more explicit (note that the +-construction is not that much more explicit, although it has a nice homotopical universal property) $\endgroup$
    – Maxime Ramzi
    Commented Nov 5, 2021 at 23:57
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    $\begingroup$ You could form the Eilenberg-MacLane space $X^{ab}= K(H_1(X,\mathbb{Z}),1)$, but this probably isn't the only choice satisfying your requirements. $\endgroup$
    – Donu Arapura
    Commented Nov 5, 2021 at 23:58
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    $\begingroup$ Here's a related question. $\endgroup$
    – Noah Snyder
    Commented Nov 6, 2021 at 0:28
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    $\begingroup$ To elaborate on Maxime's message, Quillen's plus construction is not the universal map making the fundamental group abelian, but it is the universal map making the fundamental group hypoabelian, and in fact this often produces already the abelianization $\endgroup$
    – Denis Nardin
    Commented Nov 6, 2021 at 10:30
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    $\begingroup$ The canonical example of this is taking the infinite symmetric product of X, or the free topological abelian group on X: the Dold-Thom theorem. en.m.wikipedia.org/wiki/Dold%E2%80%93Thom_theorem $\endgroup$
    – Tyler Lawson
    Commented Nov 6, 2021 at 17:42

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