# Tagged Questions

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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### orbit space of a topological manifold

Given a compact Lie group G acting freely on a topological manifold M, is it true that the orbit space M/G is also a topological manifold? If so, why?
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### If $X$ is a simplicial complex, is their a characterization of the links of the vertices of $X$ that is equivalent to the statement "$|X|$ is a manifold

We have a characterization when we want $|X|$ to be a PL-manifold, in particular that the links of all the vertices are themselves (PL) spheres. If we are in the category of PL- spaces then this is a ...
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### rational homotopy of a manifold

Given a finite dim rational homotopy type satisfying Poincaré duality, what is the best reference to when it is the rational homotopy type of a fin dim manifold?
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### Are homeomorphic open subsets of $\mathbb{R}^n$ also diffeomorphic?

Let $U_1, U_2$ be open subsets of $\mathbb{R}^n$. Both are naturally differentiable submanifold, getting the differentiable structure from $\mathbb{R}^n$. Further, both are natural topological ...
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### ANOTHER Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have another exterior differential system for one forms $U^i$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \wedge \theta^k$ for the ...
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### The geometry of crinkled aluminum foil

I wonder if the geometry of crinkled aluminum foil has been studied?            The above is a photo of foil I flattened to reuse. It might be ...
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### Status of PL topology

I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
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### What is the difference between holonomy and monodromy?

What is the difference between holonomy and monodromy? And what is the simplest example in which one is trivial and the other is not?
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### Does every vector bundle allow a finite trivialization cover?

Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$. (a) Is ...
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### When is the connected sum of manifolds orientation-independent?

Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed? If $N$ ...
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### Good overview of singularity theory

Can anyone recommend a good overview of singularity theory? In particular, quotient singularities... Thanks!
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### Triangulation of Surfaces without Jordan-Schoenflies

Does anyone know of a proof of the fact that any 2-manifold can be triangulated that does not use the Jordan-Curve Theorem or the Jordan-Schoenflies Theorem? Thanks for your help
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### When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
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### Cartan-Weil model for Equivariant Cohomology

Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$ which acts on a manifold $M$. It is quite standard that the basic forms in $\Omega^*(M) \otimes W(\mathfrak{g}^*)$ form a model for the ...
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### When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
Let $M$ be a manifold with boundary. Consider the following groups: (1) $\pi_0(\operatorname{Diff}(M,\partial M))$. (2) $\pi_0(\operatorname{Homeo}(M,\partial M))$. (3) $\pi_0(\operatorname{HomEq}(... 4answers 2k views ### Examples of non-simply connected manifolds with trivial H^1 It is known that, if a topological space is simply connected,its first homology group vanishes. The converse is not true, since for every presentation of a (say, finite) perfect group G we can ... 3answers 469 views ### symplectic form with partition on unity Assume$M$is a$2n-$dimensional differentiable manifold. Let$(U_{i})$be a open covering of$M$. With respect to this covering let$\rho_{i}$be a partition of unity. Assume that on each$U_{i}$we ... 2answers 587 views ### Cardinality of connected manifolds Consider the assertion: Every connected, but not necessarily paracompact, n-manifold is of cardinality$2^{\aleph_0}$(at least assuming the axiom of choice). For n=1 this may be proved via ... 3answers 2k views ### A “meta-mathematical principle” of MacPherson In an appendix to his notes on intersection homology and perverse sheaves, MacPherson writes Why do we want to consider only spaces$V$that admit a decomposition into manifolds? The intuitive ... 2answers 807 views ### How is the differential in complex cobordism defined? This is my first MO question...hopefully it's not a bad one... Background: As a stable homotopy theorist, I like to think of complex cobordism$MU$as a ring spectrum. If I needed to get my hands ... 3answers 399 views ### Question concerning h-cobordisms Suppose we have a cobordism$W$of manifolds$M_0$and$M_1$and suppose the inclusion of$M_0$into$W$is a homotopy equivalence. Is the same true for the inclusion of$M_1$(ie. is$W$already an h-... 2answers 538 views ### cayley transform for non-square matrices Hi, I am optimizing a function over a matrix$U$, where$U \in \mathbb{R}^{m \times n}$and$U^TU = I$. I do not want to run a constrained maximization program, since employing the constraint$U^TU = ...
Let $M$ be a connected compact manifold without boundary, $\pi:\widetilde{M}\to M$ be the universal covering map. A fundamental domain of $(\pi,\widetilde{M}, M)$ is a compact subset \$D\subset \...