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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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 ...
Chris Schommer-Pries's user avatar
16 votes
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Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups

In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M \...
Hiro Lee Tanaka's user avatar
15 votes
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Beyond smoothness-the clear picture about the notion of a differential form

In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $...
truebaran's user avatar
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Examples of manifolds with first nontrivial SW-class in degree 16 or bigger

As a module over the Steenrod algebra, $H^{\ast}(BO;\mathbb F_2) = \mathbb F_2[w_1, w_2, w_3, \dots]$ is generated by $w_{2^t}, t \geq 0$. Thus, the first nontrivial SW-class of any vector bundle $\xi ...
Jens Reinhold's user avatar
13 votes
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439 views

Structures between PL and smooth

Let $X$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $ks(X)\in H^4(X,\mathbb{Z}_2)$ is an obstruction to the existence of a PL structure on $X$. If it vanishes, ...
Philip Engel's user avatar
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11 votes
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Detecting topology change of tubular neighbourhoods via smoothness of volume function

Let $M$ be an embedded closed manifold in $\mathbb R^n$, define $M_r=\{x\in\mathbb R^n:d(x,M)<r\}$. Define $r\in\mathcal S_M$ iff $M_r\subset M_{r+\epsilon}$ is not a homotopy equivalence for all ...
Ariana's user avatar
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"The TOP h-cobordism theorem without surgery??"

Kirby and Siebenmann's book on topological manifolds contains the following intriguing passage on page 141: I believe no such proof has been discovered, though I'd be happy to be corrected on that. ...
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Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces

Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense? In other words, when does at set $S\...
Christian's user avatar
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History of the definition of smooth manifold with boundary

I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...
John Klein's user avatar
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$k$ times differentiable but not $C^k$ manifold

I asked the following question on Math Stack Exchange 3 months ago but got no answer. So maybe Math Overflow is a more suitable place for such a question: I cannot find the notion of $k$ times ...
fm12's user avatar
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Is it possible to glue together complex manifolds?

In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two ...
Kim's user avatar
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Almost Poincaré duality

Let $M^n$ be a connected, closed manifold. It has Poincaré duality with $\mathbb{Z}/2$ coefficients $H^k(M;\mathbb{Z}/2)\cong H_{n-k}(M;\mathbb{Z}/2)$, induced by cap product with the fundamental ...
Mark Grant's user avatar
8 votes
0 answers
193 views

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 (...
pureorapplied's user avatar
8 votes
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Is there a combinatorial representation of general topological manifolds similar to triangulations?

Piece-wise linear manifolds are combinatorially represented by simplicial complexes modulo Pachner moves. However, for dimensions greater than $3$, the notions of piece-wise linear and topological ...
Andi Bauer's user avatar
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Geometric argument for "easy" part of Jordan-Brouwer separation theorem without local flatness

Let $M^n \subset \mathbb{R}^{n+1}$ be an $n$-dimensional compact connected topological submanifold. The Jordan-Brouwer separation theorem says that $\mathbb{R}^{n+1} \setminus M^n$ contains two ...
Lisa's user avatar
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PL surface projections - is there a theory of folds and cusps?

For smooth surfaces, the generic singularities of a map of one surface to another are folds and cusps (Whitney). It is a standard result in singularity theory that the generic isotopy of such a map is ...
johnwbarrett's user avatar
7 votes
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288 views

Can every finitely presented group be realized as a fundamental group of a compact four-dimensional smooth submanifold $\mathbb{R}^4$?

Every finitely presented group is realized as a fundamental group of a two-dimensional complex (a simple exercise on Van Kampen's theorem). I was told that a two-dimensional complex can be well ...
Arshak Aivazian's user avatar
7 votes
0 answers
279 views

When does the tangent microbundle of a closed orientable topological $4k$-manifold have a trivial rank 2 subbundle?

$\DeclareMathOperator{\Top}{Top} \DeclareMathOperator{\co}{H}$Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known (David Frank, On the index of ...
Cihan's user avatar
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7 votes
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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 $...
Paris's user avatar
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How much differs the category of real-analytic manifolds from $C^\infty$ ones?

I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
John117's user avatar
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227 views

GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
wonderich's user avatar
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7 votes
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Differential topology on arbitrary fields

What do the differential topology theories on arbitrary fields have in common? Different differential topology theories There is "ordinary" differential topology on real manifolds, with its rich ...
Manuel Bärenz's user avatar
7 votes
0 answers
288 views

Smooth dependence on parameters of invariant manifolds

This may be considered a followup question to smooth dependence of stable manifold on parameters. In particular, I would like to know in more detail how smooth parameter dependence of a (generalized) ...
Jaap Eldering's user avatar
7 votes
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427 views

Reference Request: Topological h-cobordism theorem in higher dimensions

I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so. The h-cobordism theorem is true in the topological and in the smooth category in ...
Tina's user avatar
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645 views

Homometric $\Rightarrow$ isometric?

Suppose you know that there is a mapping between two Riemmanian manifolds $M_1$ and $M_2$ such that, for each $x_1 \in M_1$, the (codimension-1) measure of the set of points at distance $d$ from $x_1$ ...
Joseph O'Rourke's user avatar
7 votes
0 answers
176 views

a "homological dimension" for embedding of manifolds

Let $A\to B$ be a surjective map of commutative $k$-algebras, and suppose $C\to B$ is a free resolution of $B$ as an $A$-algebra, meaning that $C$ is a free non-negatively graded commutative $A$-...
Dima Roytenberg's user avatar
6 votes
0 answers
123 views

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 ...
geooranalysis's user avatar
6 votes
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571 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. ...
Arshak Aivazian's user avatar
6 votes
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207 views

If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ be a triangulable manifold?

If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ always be a triangulable manifold? Let $X_d$ be a $d$-manifold which is NOT a ...
wonderich's user avatar
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6 votes
0 answers
239 views

Signature of a non-compact manifold

Let $v_0,\dots,v_n\in\mathbb{Z}^2$ be integer vectors which satisfy the condition $\det\begin{pmatrix}v_{k-1}&v_k\end{pmatrix}=(-1)^k$, whose relevance will become apparent in a moment. We may ...
user156275's user avatar
6 votes
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174 views

Reference request: Existence and regularity for parabolic PDEs with smooth coefficients on compact manifolds with boundary

I'm looking for a reference for a statement like: Let $M$ be a $n$-dimensional smooth compact manifold with smooth boundary $\partial M$. In coordinates, let $\mathcal L$ have the form $\mathcal L ...
Nathanael Schilling's user avatar
6 votes
0 answers
115 views

Topological constraints for existing of certain differential operators on manifolds

At the beginning a word of warning: this would be rather vague question: vague as it is, I'm not requiring a precise answer, rather some intuitive explanation. In the flat case $M=\mathbb{R}^n$ ...
truebaran's user avatar
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6 votes
0 answers
380 views

Do all topological manifolds admit locally flat embeddings into R^n?

In his 1969 paper “Locally flat imbeddings of topological manifolds” Lees proved that a closed oriented second countable topological manifold admits a locally flat embedding into some R^n. Does the ...
Dmitri Pavlov's user avatar
6 votes
0 answers
168 views

What are the regular monomorphisms and regular epimorphisms of the category of smooth premanifolds?

A premanifold is a locally ringed space locally isomorphic to an open subset of Euclidean space equipped with its sheaf of smooth functions. No assumption of paracompactness or the Hausdorff property. ...
Arrow's user avatar
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6 votes
0 answers
306 views

Embedding tower in low codimension

If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm{holim} T_k F$. The ...
Ben Knudsen's user avatar
5 votes
0 answers
153 views

The fundamental group of the complement of badly embedded open $n$-ball in $\Bbb R^n$

Let $\mathcal D^n$ be an open subset of $\Bbb R^n$ such that $\mathcal D^n$ is homeomorphic to $\{x\in \Bbb R^n:|x|<1\}$. Suppose $\Bbb R^n\setminus \mathcal D^n$ is path-connected. How bad can $\...
Random's user avatar
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0 answers
195 views

Does the (Poincare) dual complex represent the same topology?

To start with, consider some abstract $3$-dimensional simplicial complex $\Delta$ representing a manifold without boundary, for simplicity. Then, there is this well-known construction of the "(...
B.Hueber's user avatar
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5 votes
0 answers
292 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 deduced from fairly intricate topological results of ...
Arrow's user avatar
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5 votes
0 answers
107 views

Induced new structures on Poincare dual manifolds

"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...
wonderich's user avatar
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5 votes
0 answers
160 views

Theoretical justification of time-series forecasting using Takens' embedding

This is a cross-posting where I couldn't get an answer. In the meantime I have tried to improve the original logic: As in Takens original paper about his embedding theorem, consider a compact $m$-...
Sarem Seitz's user avatar
5 votes
0 answers
103 views

On the embedding of manifolds into infinite-dimensional spaces

Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
erz's user avatar
  • 5,385
5 votes
0 answers
463 views

What is the dimension of $M/G$ if it is a manifold and $G$ acts freely and smoothly?

Let $G$ be a Lie group acting smoothly and freely on a smooth manifold $M$. Suppose that the quotient space $M/G$ is a topological manifold. Do we have $$\dim(M/G)=\dim M-\dim G?$$ Notes: This ...
SHP's user avatar
  • 779
5 votes
0 answers
184 views

A question about something like "shelling" in a PL manifold

If $P$ and $Q$ are compact codimension zero submanifolds of a PL manifold, say that they meet nicely if $P\cap Q$ is a codimension zero submanifold of both $\partial P$ and $\partial Q$. In ...
Tom Goodwillie's user avatar
5 votes
0 answers
195 views

Is there a Whitney-type theorem Cauchy manifolds?

Let $M$ be a Cauchy space whose induced topological space is a second-countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^m$. Does it follow that there exists a subspace $N$ of $\...
user avatar
5 votes
1 answer
208 views

Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have. Is there a non-separable complex manifold? Can a non-separable complex ...
Joseph Van Name's user avatar
4 votes
0 answers
218 views

To what extent is the Nash embedding not unique?

Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique? It is clear that the set of all such embeddings ...
dennis's user avatar
  • 423
4 votes
0 answers
60 views

Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
Spencer Kraisler's user avatar
4 votes
0 answers
351 views

Obstruction of smooth structure

The first 24 lectures of Jacob Lurie on Geometric Topology [1] gave a concise introduction to the comparison of smooth manifolds and piecewise-linear manifold. In the first five lectures, it is shown ...
Student's user avatar
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4 votes
0 answers
175 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)} \|\...
Matheus Andrade's user avatar
4 votes
0 answers
159 views

(Non-)Orientability of non-triangulable manifolds

We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least ...
annie marie cœur's user avatar