Questions tagged [differential-topology]
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
1,741
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What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?
H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, (1989), p. xi:
...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic ...
2
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0
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91
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If a Compact $n$-Manifold Immerses in $\mathbb{R}^{n+1}$ is there a Locally Flat Immersion?
Suppose that $M$ is a compact, topological $n$-manifold and there is a topological immersion (i.e. local embedding) of $M$ into $\mathbb{R}^{n+1}$. Is there necessarily a locally flat immersion of $M$...
1
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0
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237
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Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle
Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be an oriented $m$-dim ball bundle, i.e. each fiber is an oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\...
2
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1
answer
123
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Gluing isotopic smoothings
Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
1
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0
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126
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Middle Betti number bound of special $4$-manifolds
I was thinking of the following question regarding $4$-manifolds as follows.
Let $M$ be a compact, oriented, smooth $4$-manifold with a smooth connected boundary, say $N$. Let $M$ be simply connected ...
6
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0
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123
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A particular case of the general converse to the preimage (submanifold) theorem
I was thinking whether it would be possible to develop a converse to the preimage theorem in differential topology and I found the following post:
When is a submanifold of $\mathbf R^n$ given by ...
8
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0
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193
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A modified version of the converse to the Sard's Theorem
When I learned Sard's Theorem in differential topology by myself, I was thinking whether it would be possible to prove a converse version of the theorem. That is to say, can we somehow show that each (...
1
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0
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200
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Question regarding affine fibre bundles
Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...
6
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2
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431
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The convex hull of a manifold whose cobordism class is trivial
Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...
12
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1
answer
298
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Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?
In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
16
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4
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1k
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Can one glue De Rham cohomology classes on a differential manifolds?
Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k_{...
16
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3
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1k
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Converse to Hopf degree theorem
Below, I mean smooth oriented closed connected manifolds and smooth maps (but am happy to hear about the topological category, or unoriented manifolds, etc instead).
Say that $X^n$ has the Hopf ...
1
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0
answers
30
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Results on compact slices in a regular foliation
Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...
6
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1
answer
513
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Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$?
Let $M$ be a simply connected, (orientable), non-compact, 3-manifold without boundary. Must $M$ be homeomorphic with a topological subspace of $\mathbb{R}^3$?
3
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1
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323
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How to show that $\text{Man}(M,\mathbb{R}^n)\cong \mathbb{R}\text{-Alg}(C^\infty(\mathbb{R}^n),C^\infty(M))$?
I'm trying to show that manifolds are affine, i.e. $\text{Man}(M,N)\cong \mathbb{R}\text{-Alg}(C^\infty(N),C^\infty(M)) $. If I could show this for $N=\mathbb{R}^n$, then I know how to do the rest ...
2
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0
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116
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Question about spin map
I'm confused with the following definition of a spin map.
A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...
1
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0
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108
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Existence of a local spinor bundle
I am confused about the existence of a local spinor bundle.
My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...
30
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1
answer
1k
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Are homeomorphic representations isomorphic?
Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are ...
4
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2
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Representation of fundamental group and flat connections
I read Differential Geometry Of Complex Vector Bundles by Kobayashi, and he says there that a vector bundle $E$ has flat connection is equivalent to $E$ being defined by a representation of $\pi_1$. ...
2
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1
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298
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Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely
Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...
5
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1
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Is the smooth singular simplicial set of a smooth manifold a Kan complex?
It is classical that the singular simplicial set of a topological space is a Kan complex. This is elementary and already due to presumably Kan.
Q: Is the smooth singular simplicial set of a smooth ...
4
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1
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240
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A cobordism theory from Hirsch's "Differential Topology" (reference request)
The following is exercise 5 on p. 176 in Hirsch's "Differential Topology" (corrected 6th printing):
Let $\eta = (p,E,B)$ be a fixed vector bundle over a compact manifold $B$, $\partial B = \...
3
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0
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162
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The boundary of the transversal pre-image of a submanifold with boundary
A similar question on MSE without answer.
Let $M, N$ be smooth manifolds such that $\partial N=\varnothing$. Let $A$ be a smoothly embedded submanifold of $N$ such that $\partial A\neq \varnothing$. ...
9
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2
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338
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Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds
This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following ...
2
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0
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81
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Gysin homomorphism of an inclusion to Kähler tubular neighborhood
Let $Z\subset U$ be a Kähler tubular neighborhood of a compact manifold $Z$ of codimension $r$.
Consider de Rham complexes of smooth differential forms $\Lambda^{*,*}(Z),\Lambda^{*,*}(U)$, let $\...
6
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0
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161
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Is the category of diffeological spaces a full subcategory of locally ringed spaces?
It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here).
Is a similar ...
1
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1
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134
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Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood
A similar post on MSE without answer.
Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold of $M$ of co-...
0
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0
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84
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The closure of the subgroup generated by a vector field may not be compact
Suppose $X$ is a vector field on a manifold $M$, consider the one parameter group:
$$L=\left\{\phi^t_X: t\in\mathbb{R}\right\}$$
where $\phi^t_X$ is the flow of the vector field $X$, which sends $p\in ...
1
vote
1
answer
147
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Uniqueness of "stretching" (subject to constraints) for a two-dimensional figure
The New York Times, reporting on Dennis Sullivan's Abel prize, recounts the incident that lured Sullivan from chemical engineering to mathematics:
One day during an advanced calculus lecture, the ...
11
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1
answer
846
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Intuition/meaning behind/physical content of the concept of a smooth structure
Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological ...
2
votes
2
answers
175
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An equivalence relation on knots similar to concordance
Let $L_1$ and $L_2$ be two nonintersecting picewise-linear or smooth knots in $\mathbb R^3$. Suppose they are ambient isotopic. Does there exist an embedded surface $f: S^1\times[0,1]\to \mathbb R^3$ ...
0
votes
1
answer
334
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Sheaf of Kähler differentials for complex manifold
Let $X \subset \mathbb{C}^n$ be an analytic set which means that it is the zero locus of holomorphic functions $f_1,f_2,\dotsc,f_n$ on $\mathbb{C}^n$ and suppose that there is a singular locus ...
13
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2
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562
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When are bundles of odd and even differential forms isomorphic?
Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ ...
5
votes
1
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710
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Kirby diagrams: sliding 1-handles over 1-handles and ribbon disks
Consider the Kirby diagram $ D$ given by a 2-component unlink, both dotted circles.
In general, when performing a 1-handle slide over another 1-handle, the band chosen must not link any dotted circle,...
2
votes
1
answer
236
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Quantitative results for stabilizing tangent bundles of homology spheres
I'll begin with a broad question: if $M$ is a smooth manifold and $E \to M$ is a stably trivial bundle, can one determine lower bounds on the rank $k$ of the trivial bundle needed such that $E \oplus \...
2
votes
1
answer
232
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Equidistant points on a compact Riemannian manifold
Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows:
$K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at ...
5
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0
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154
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Potential theory as a tool in extrinsic flows
Let $M \subseteq \mathbb{R}^n$ be a submanifold. For a point $x$ disjoint from $M$, we can define the electric potential $\Phi(x) = \int_M \frac{dM}{|x-m|^{n-2}}$, which is smooth and harmonic where ...
0
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0
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89
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Topologies in $\mathcal{C}^\infty(M,N)$
Naively, one could topologise the set of smooth (ie $\mathcal{C}^\infty$) maps between two smooth manifolds $M \to N$ with the subspace topology $\mathcal{C}^\infty(M,N) \subseteq \mathcal{C}^0(M,N)$, ...
11
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2
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554
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Homotopy properties of Lie groups
Let $G$ be a real connected Lie group. I am interested in its special homotopy properties, which distinguish it from other smooth manifolds
For example
$G$ is homotopy equivalent to a smooth compact ...
6
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0
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154
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Uniqueness of normal microbundle of a smooth embedding
Suppose $M$ is a topological manifold and $\iota: N\hookrightarrow M$ be a submanifold. A normal microbundle of $N$ consists of an open neighborhood $U$ of $N$ and a retraction $\pi: U \to N$ such ...
5
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2
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242
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Patching up two trivial fibre bundles induces homology equivalence
I was wondering to ask this question may be it's a silly one. I could not prove or disprove it.
Let $X,Y$ be smooth connected manifolds. Let $X=X_1\cup X_2$ ($X_i$'s sub-manifold of $X$) and $X_1 \cap ...
9
votes
1
answer
501
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Submersion vs fiber bundle
If one starts with a fiber bundle $f: X \to Y$ so that fibers having trivial integral homology by using spectral sequence one can get the induced map $f_*: H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is ...
3
votes
2
answers
300
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Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$
$\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$.
I would like ...
4
votes
1
answer
331
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Poincaré–Bendixson Theorem on a compact, connected, orientable, two-dimensional manifold
I'm currently reading the article "A Generalization of a Poincaré–Bendixson Theorem to Closed Two-Dimensional Manifolds" by Arthur Shwartz. The paper first establishes a result for minimal ...
3
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0
answers
187
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Proper isotopy of proper embeddings of manifolds
We know from Theorem 2.2 in(http://www.map.mpim-bonn.mpg.de/Embeddings_of_manifolds_with_boundary:_classification#6.2) that
Any two smooth embeddings of closed oriented $n$-manifold(n>1) in $\...
6
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0
answers
241
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Regarding homology of fiber bundle
Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...
15
votes
2
answers
1k
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Diffeomorphism group of the projective plane
First of all, I am interested in the general case of a non-orientable manifold but let's for now consider the projective plane $\mathbb{R}P^2.$ In short, I am curious if there is any relation between ...
3
votes
0
answers
224
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Standard sutured (?) Heegaard splitting
I am trying to make sense of what is going on in [Cas16] in terms of diagrams. Let me sum up the construction a bit, where $n\leqslant k$ are integers and $b\geqslant 1$ as well.
$C_{k,b,n}$ denotes ...
5
votes
0
answers
90
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Equivariant imbedding of compact manifold
Let $G$ be a compact Lie group smoothly acting on a smooth compact manifold $X$.
Is it true that there exists a smooth $G$-equivariant imbedding of $X$ into a Euclidean space acted linearly (and ...
9
votes
1
answer
706
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Is there a version of the Poincaré–Hopf theorem for manifold with corners?
As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...