TITLE: Retraction onto a circle in a simplicial complex
QUESTION [8 upvotes]: Let $X$ be a connected space homeomorphic to a finite simplicial complex. If there is an embedding $i: S^1 \hookrightarrow X$ which has a retract $r: X \rightarrow S^1$, then necessarily the first Betti number $b_1(X)$ is nonzero. Is the condition $b_1(X) \neq 0$ sufficient for a circle retract to exist?

REPLY [4 votes]: As mentioned in Cihan's answer, this is a result of Borsuk. The proof is divided into two papers (see [1, Théorème 30] and [2, Korollar 11]).
A proof of this result (for Peano continua and rather difficult to read) appears in Kuratowski's book [3, S 57, III, Theorem 4].
In Section 1.1.3 (pages 8-11) of my thesis [4], I provide a proof for arbitrary (not necessarily finite) simplicial complexes which is based on Kuratowski's proof.
I sketch here a different way to prove this result for a finite simplicial complex $X$. First we take a 1-cycle in $Z_1(X)$ which represents a generator of a direct summand of $H_1(X)$ isomorphic to $\mathbb{Z}$. Then we need to take a representative of this cycle which is a simple closed curve. If $X$ is not a surface and $X$ has no local separating points (a point $x\in X$ is a local separating point if there is a connected open neighborhood $U\ni x$ such that $U-x$ is disconnected) it is possible to do this. The main idea is to consider three 2-simplices with a common edge and to use this to remove intersections one by one. It is easy to reduce to the case in which $X$ has no local separating points. Finally, surfaces are managed using the classification.
In every proof I know, it is necessary to subdivide the complex at some point. I do not know if it is always possible to find a subcomplex of the original complex homeomorphic to $S^1$ which is a retract.
[1] K. Borsuk. Quelques théorèmes sur les ensembles unicohérents, Fund. Math. 17 (1931), no. 1, 171–209.
[2] K. Borsuk. Über die Abbildungen der metrischen kompakten Räume auf die Kreislinie, Fund. Math. 20 (1933), no. 1, 224–231.
[3] K. Kuratowski. Topology vol. ii, Academic Press, 1968.
[4] I. Sadofschi Costa. Fixed points of maps and actions on 2-complexes, PhD thesis, Universidad de Buenos Aires, 2019. Available at http://cms.dm.uba.ar/academico/carreras/doctorado/tesisSadofschi.pdf.