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# Questions tagged [manifolds]

A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

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Because $TM$ is an embedded $2m$-dimensional submanifold of $T\Bbb R^n\approx \Bbb R^n\times \Bbb R^n$, we can define the smooth map $\varphi: TM\to \Bbb R$ by $(x_1,\cdots,x_n,v_1,\cdots,v_n)\... 1answer 184 views ### Existence of universal arrow from manifolds to forgetful functor of Lie groups Let$M$be a manifold, and$U$be the forgetful functor from the category of Lie groups to the category of manifolds. My question is whether there is a universal arrow$(G, i)$from$M$to$U$? More ... 2answers 206 views ### Line bundles trivial outside of codimension 3 Let$X$be a CW complex (possibly a topological/smooth manifold) of dimension$n$,$L\to X$a complex line bundle and$Y\subset X$a subcomplex (possibly a submanifold) contained in the codimension 3 ... 0answers 132 views ### Category of Manifolds and Maps: TOP$\supseteq$TRI$\supseteq$PL$\supseteq$DIFF? [closed] Please let me denote the following (TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold (PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF (PL) ... 1answer 703 views ### Cobordism Theory of Topological Manifolds Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds. Cobordism Theory for DIFF/Differentiable/smooth manifolds However, there are ... 0answers 104 views ### Naturality of Poincaré–Lefschetz Let$X$be compact and Hausdorff,$A\subseteq B\subseteq X$both closed such that$X\setminus A$is an open orientable$d$-manifold. Then also$X\setminus B$is an open orientable$d$-manifold. We ... 0answers 69 views ### How do these topological results imply the inverse function theorem? In this MO question, Terrence Tao inquires about the everywhere differentiable inverse function theorem. This answer claims the theorem may be deduce from fairly intricate topological results of ... 0answers 72 views ### “Smooth” Serre Fibrations (?) Let$M,N$be manifolds,$f:M \to N$be a map. In order to understand if$f$is a serre fibration, it is enough to test it against differntiable maps$I^p \to M, I^{p+1} \to N$? What about smooth maps?... 0answers 123 views ### Counting fixed points for Hamiltonian symplectomorphisms on$T^{2}$This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in ... 0answers 79 views ### How much is the theory of piecewise-smooth manifolds different from the PL and smooth ones? There are many differences between the smooth manifolds category and piecewise linear manifolds category when it comes to classification, embedding of these manifolds inside each other or otherwise. ... 0answers 75 views ### Notion of knotting for a compact$n$-manifold inside an$m$-manifold (with the constraint$n=m?2$) What is the notion of knotting for a compact$n$-manifold inside an$m$-manifold? 1answer 90 views ### Unknotting a compact manifold in the PL setting The general position theorem asserts that any$Mm$-manifold unknots in$R^n$provided$n\geq 2m+2$. The general position theorem assumes a smooth setting. Is unknotting still hold in the PL setting?... 2answers 202 views ### Notion of linking between two general$p$and$q$manifolds embedded in a higher dimensional manifold Let$M$,$N$be compact, connected, oriented manifolds without boundary embedded in$\mathbb{R}^m$of dimensions$p$and$q$respectively. I know that when$m=p+q+1$we can define the linking between$...
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### Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere

Is there an embedding (i.e. injective continuous map) $$\phi:\Bbb R P^2\hookrightarrow S^4\subseteq\Bbb R^5$$ of the projective plane $\Bbb R P^2$ into the $4$-sphere, that is transitive, i.e. ...
1answer
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### Closed manifold with non-vanishing homotopy groups and vanishing homology groups

Is there a closed connected $n$-dimensional topological manifold $M$ ($n\geq 2$) such that $\pi_i(M)\neq 0$ for all $i>0$ and $H_i(M, \mathbb{Z})=0$ for $i\neq 0$, $n$? The manifold $S^1\times S^2$ ...

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