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If $A\subset \Bbb R^2$ then is the following statement true? $\{(x,y)\in {(A\times A)/ \sim}\,\,\,|\,\, (x,y)\sim(y,x)\}\simeq$ Möbius strip $\iff A$ is a Jordan curve.

Let $M$ denotes the Möbius strip. Then is it true that For every continuous map $f:M\to M$ there is $x\in M^\circ$ ($x\notin\partial M$) such that $f(f(x))=x$?

This is a cross post of MSE. Q1: What does "irreducible manifold" mean (not definition)? My understanding of "irreducible manifold" is "is not reducible (homotopic or deformation or homeomorph or be …

Here is the comment to this book in author's web page: Differential Forms in Algebraic Topology (with Raoul Bott), third corrected printing, Graduate Text in Mathematics, Springer, New York, 1995. …

Summary of comments and other sources There are at least 4 similar concepts: Irreducible smooth manifold: As Ryan Budney said, "Regarding high dimensions, generally irreducible manifolds do not exist …

Response to the first question: Pantilie, Radu, A simple proof of the de Rham decomposition theorem, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 36, No. 3-4, 341-343 (1992). ZBL0811.53040.

Today I read about Gromov's definition of minimal volume for smooth manifolds. $$\min {\rm Vol}(M):=\inf_{|K_g|\leq1}\{{\rm Vol}(M,g)\}.$$ Gromov's conjecture states that for every closed simply con …