# 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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A continuous map $f$ between two metric spaces is said to be a $r$-map if preimage of each point under $f$ has diameter atmost $r$. Suppose $D^n=\{x\in \mathbb{R}^n\mid ||x||\leq 1\}\subset \mathbb{R}^... 5 votes 0 answers 356 views ### Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary? Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary? I am also interested in several variations of this question. ... 2 votes 1 answer 144 views ### A Riemann surface is automatically paracompact [A question I remember from many years ago.] Definition A Riemann surface is a connected complex manifold$X$of complex dimension one. This means that$X$is a connected Hausdorff space that is ... 3 votes 0 answers 72 views ### non-smooth manifold with circle action (with fixed points) I am interested to know if there a non-smooth manifold (i.e. a closed topological manifold admitting no smooth structure)$M$, having a continuous action$M \times S^1 \rightarrow M$, and the number ... 3 votes 1 answer 149 views ### Open manifolds which have stable$\pi_1$at infinity but are not inward tame Let$M$be a$1$-ended open manifold. An important result of Siebenmann states that (in dimension$\geq6$) if$M$is$(i)$inward tame, i.e. for every closed neighborhood of infinity there exists a ... 11 votes 2 answers 626 views ### Can we embed a closed manifold into a homotopy equivalent CW complex? Suppose$X$is a CW complex and$M$is a closed manifold and suppose further that there exists a homotopy equivalence$X \simeq M$. Does there exists an embedding of$M$into$X$(i.e. an injective (... 4 votes 0 answers 134 views ### Classifying singularities of the Ricci flow Context: A solution$(M^n, g(t))$of the Ricci flow is said to encounter a Type III Singularity if$g(t)$is defined for all$t \geq 0$and: $$\sup _{\mathcal{M}^{n} \times[0, \infty)} \|\... 12 votes 1 answer 436 views ### Are there examples of Einstein manifolds with unbounded curvature? Are there any known examples of Einstein manifolds (M, g) such that$$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$I'm looking for these examples because they might provide a counter-example to a ... 12 votes 1 answer 383 views ### Roadmap for L-Theory Background: I spent sometime reading about algebraic K-theory and started reading research papers on the subject with relative facility at least I do understand constructions, statements of the ... 7 votes 0 answers 144 views ### Homotopy equivalent cartesian product of closed manifold I'm little bit lost with the following question: I have four connected closed orientable manifolds M,N,S,S' such that S and S' are homotopy equivalent and M\times S is homotopy equivalent to ... 22 votes 0 answers 585 views ### Do most manifolds have symmetries? or not? Let us say that a (closed, connected) manifold has a symmetry if it admits a non-trivial action by a finite group. Note that I am not asking the action to be free. So for example rotating the 2-sphere ... 6 votes 3 answers 460 views ### Contractible set in a manifold Let M be an n-dimensional topological closed manifold. Suppose K is a compact subset of M which is contractible in the sense that there exists a continuous map F:K \times [0,1] \to M with F(... 9 votes 2 answers 382 views ### Is \operatorname{Spin}(8) a direct product of \operatorname{Spin}(7) and S^7? Is \textrm{Spin}(8) a direct product of \textrm{Spin}(7) and S^7? I met this statement in the literature, but without a reference. If it is true, where is it explicitly written? 0 votes 0 answers 64 views ### Hypothesis on W to achieve d(x,F(W))<\sum\epsilon_n Writing a paper, I'm trying to formulate the following technical result: Let X be a manifold, W\subset\subset X. Let f_k:U_k\to X be continuous, where U_k\subset X is an open neighborhood of ... 10 votes 2 answers 513 views ### Do colimits of manifolds coincide with underlying colimits as topological spaces? Categories of manifolds (possibly with extra structure) tend not to have all colimits. Other questions have addressed when colimits of manifolds exist. I'd like to know what we can say in general ... 8 votes 1 answer 552 views ### When is a triangulation of sphere two-colorable? Let T be a triangulation of sphere. We say that T is k-colorable if the triangles of T can be assigned with k colors such that any two triangles with a common edge have different colors. I ... 2 votes 0 answers 105 views ### Birth of chaos due to nonautonomous perturbation Let \sigma, b>0. I want to study the dynamics of the map$$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$such that$$T_{\sigma,b}(n,\theta,y) = (\... 3 votes 2 answers 130 views ### Embedded submanifold in a cylinder Let$M^n$be an$n$-dimensional topological closed manifold. Suppose there exists an embedding$i:M \to M \times [0,1]$such that$i(M)$is contained in the interior of$M \times [0,1]$and separates$...
Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?