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According to David Roberts comment and the following paper it is open for dimensions $n\geq 4$. Pan, Jiayin, A proof of Milnor conjecture in dimension 3, J. Reine Angew. Math. 758, 253-260 (2020). ZBL …

Gluing two Mobius strips along their edges is a Klein bottle.

Cross post from MSE. and sorry if this is an obvious question. Here is a line of proof of Theorem 1.15 from Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. Prov …

If I am not mistaken this has been done by R. Lakshminarayanan and you can find it in Bródy, F. (ed.); Vámos, T. (ed.), The Neumann compendium, World Scientific Series in 20th Century Mathematics. 1. …

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. …

Ben Weinkove 5 lectures The Kähler–Ricci flow on compact Kähler manifolds which has been collected in Bray, Hubert L. (ed.); Galloway, Greg (ed.); Mazzeo, Rafe (ed.); Sesum, Natasa (ed.), Geometric an …

I want to know whether there are ways to use algebraic methods for solving graph theory problems (graph coloring problems). For example, is it possible to prove the four-color theorem purely with alg …

Let $(M^n,g)$ be a Riemannian manifold that admit a unit Killing vector field $X$. i.e., $\mathscr{L}_Xg=0$. Is it possible that there exist a smooth function $f$ on $M$ such that $X=\mathrm{grad}f$? …

In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces. Is this a valid theorem? Any reference t …

As Igor Belegradek commented, the correct statement is as follows: Theorem (classification of closed simply connected manifold with nonnegative curvature operator): A closed simply connected manifold …

An Introduction to Lie Groups and the Geometry of Homogeneous Spaces written by By Andreas Arvanitogeōrgos is a useful reference Read online here. Another good reference for physic and math students …

This is a cross-post from MSE. These four screenshots from milnor's book baffled me a bit (pages 24, 50, 51 and i-iii resp.): In first one, there is no Theorem 3.1 in the book, but there i …

This is a cross post of MSE post somehow: Is there any example of compact fiber bundle $E$ with noncompact fibers $F$? Obviously if the base space $B$ is $T_1$ then there is no such example.

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 …

This book must be useful: An Introduction to Differentiable Manifolds and Riemannian Geometry written by William Munger Boothby Page 319. Read online in Google Link.