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”.

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2
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1answer
117 views

Ehresmann fibration theorem for manifolds with boundary

All manifolds in consideration may have nonempty boundary and may be disconnected. Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...
6
votes
2answers
234 views

Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question. I can think of at least four topologies to put on $C_c(M)$: Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...
1
vote
0answers
49 views

freedom in choosing a smooth function of compact support [migrated]

Suppose $\Omega$ represents a bounded domain in $\mathbb{R}^n$. For ease, let $\Omega$ be a ball around the origin. Let $\varphi$ be a smooth radial function defined on $\Omega$. I am trying to ...
0
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0answers
72 views

Does a (non-closed) differential 1-form define a curve? [closed]

I am trying to understand under what conditions the following procedure properly defines a curve. Take a manifold $M$ with a (non-closed) 1-form $B$ and an exact 1-form $dA$. Define the functions ...
5
votes
1answer
218 views

When does the Borel construction have the homotopy type of a CW-complex?

Suppose that $G$ is a Lie group acting smoothly on a manifold $M,$ does the Borel $M \times_G EG$ construction have the homotopy type of a CW-complex? If not, under what conditions would this be true? ...
1
vote
1answer
74 views

Are normal deformations of an embedding open in the $C^{\infty}$-space of embeddings of a compact smooth manifold`

Let $M$ be a compact smooth manifold (closed, for simplicity), $n\in\mathbb{N}$ and equip the space of embeddings $Emb(M,\mathbb{R^n})$ wih the Whitney-$C^{\infty}$-topology. (The weak and the strong ...
2
votes
0answers
59 views

Analogue of the Euler class of a circle bundle and the global angular form

This is a general question that asks whether there is geometric significance to differential forms that restrict to G-invariant volume elements on the fibers of a principal G-bundle. For an SO(2) ...
7
votes
2answers
592 views

Stokes theorem with corners

I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19: Notation: for $1 \le n \le m$ $\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ ...
2
votes
1answer
125 views

References on the Free Loop Space

I intend to approach the paper of Wolfgang Ziller: "The Free Loop Space of Globally Symmetric Spaces", but I need the proper background on the foundations of the study of Free Loop Spaces. I obtained ...
8
votes
2answers
214 views

A kind of uniqueness for the double of a manifold

Given two smooth, connected manifolds, M, N, with the same boundary. If their doubles, D(M) and D(N) are diffeomorphic, does it follow that M and N are diffeomorphic ? The condition on the boundary is ...
5
votes
0answers
141 views

proving the injectivity half of de Rham's theorem by construction in degrees other than $1$ and $n$

(This is a revision of a question I asked on MSE.) Let $M$ be a smooth manifold of dimension $n$, and let $\omega$ be a differential form of degree $p$ on $M$. Then we have (I'm pretty sure) the ...
1
vote
1answer
81 views

Why does the Gluck twist on a spun knot give the standard $S^4$?

Given a knotted arc $A \subset D^3$ (whose endpoints are, say, at $(\pm 1,0,0)$), the spun knot on this arc is $$\partial\left((D^3, A) \times D^2\right), = (\partial(D^3,A) \times D^2) \cup ((D^3,A) ...
1
vote
0answers
99 views

Estimates of eigenvalues of elliptic operators on compact manifolds

The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula ...
1
vote
0answers
104 views

Classification of $SU(n)$-principal bundles over a four-dimensional base

It is well-known that a principal $SU(2)$-bundle $P$ over a four-dimensional manifold $M$ is topologically classified by its second Chern-class $c_{2}(P)\in H^{4}(M,\mathbb{Z})$, as explained for ...
1
vote
1answer
105 views

Boundary components of a subsurface

Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many ...
5
votes
0answers
190 views

Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n. Is there a citeable reference for this result? For the sake of being definite, let's say that “citeable” means ...
-2
votes
1answer
170 views

Degree of a rational function [closed]

I would like to have a simple proof for the following result: Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...
2
votes
1answer
111 views

Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E (i.e. line bundles). Let $$ n_1:= ...
0
votes
1answer
197 views

On Gromov's Theorem on Symplectic Homotopy

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...
1
vote
0answers
30 views

Existence of triangulation of Lipschitz domains

Consider a bounded Lipschitz domain $\Omega \subset \mathbb R^n$. Q1: Can its closure $\overline\Omega$ be triangulated? Q2: If yes, can the triangulation be chosen as finite? Furthermore, how ...
1
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1answer
187 views

Linearisation of Einstein operator

Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. The Ricci curvature can be viewed as a differential operator ...
2
votes
1answer
146 views

Lorentzian metrics on the torus up to continuos deformations

Any two Riemannian metrics can easily be deformed into each other, only obtaining positive definite metrics in between. However, for metrics of other signatures this might not be possible. Which ...
5
votes
2answers
159 views

Is the strong Whitney topology connected?

$\def\bbR{\mathbb R}\def\ssp{\kern.4mm}$Put more precisely, let $C$ be the set of all continuous functions $f:\bbR\to\bbR$ when $\bbR$ has its standard order topology. Let $\mathscr T$ be the set of ...
6
votes
1answer
200 views

Is there an analog of compactified moduli spaces(/stacks) for smooth manifolds?

This question has been inspired by an answer to the question Reference request: Topology on the space of smooth compact submanifolds; I've asked it in a comment to that answer but then decided to make ...
4
votes
0answers
126 views

When does a manifold 'deformation retract' into a small neighborhood of some k-dimensional subpolyhedron?

Under which conditions does a m-manifold $M^m$ admit a deformation retraction to a small neighborhood of some k-dimensional subpolyhedron? Or, under which conditions is the identity map $id_M$ of a ...
1
vote
0answers
66 views

Global topological equivalence of Morse functions

Two Morse functions $f$ and $g$ are called topologicaly equivalent if there are diffeomorphism $h$ of the source and orientational-preserving diffeomorphism $k$ of the target such that $f=k\circ ...
0
votes
1answer
122 views

Gluing submanifolds along their common boundary

This might be too elementary for this site, but I asked first on math.stackexchange and didn't get an answer even after offering 250 bounty points. Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ ...
2
votes
1answer
263 views

Gauss-Bonnet invariant Ω: explicit intrinsic expression for Π in Ω=dΠ?

Let the Gauss-Bonnet form be $\Omega\propto\text{Pf}(\Omega^i{}_j)$ with $\Omega^i{}_j$ the curvature 2-form of an even-dimensional manifold with dim=$n$. The Gauss-Bonnet form is exact, as shown in ...
0
votes
1answer
131 views

Ambient isotopy of the diagonal submanifold in product space

Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold ...
9
votes
1answer
328 views

Reference request: Topology on the space of smooth compact submanifolds

In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...
4
votes
0answers
88 views

Connected sum of chiral manifolds

Let $M,N$ be two closed, smooth, orientable manifolds of the same dimension and assume that these manifolds are chiral, i.e. they do not admit an orientation reversing automorphism. Then there are two ...
3
votes
0answers
88 views

Multiplicativity of combinatorial l classes

For closed smooth manifolds $M$ and $N$, the Hirzebruch $L$ class is multiplicative, i.e. $L(M\times N)=L(M)L(N)$. Is this property still true if $M$ and $N$ are assumed to be closed topological ...
2
votes
0answers
117 views

Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...
0
votes
0answers
69 views

What does deg=0 imply for the gauss map?

I am facing the following problem. I have a Riemannian manifold $(M,g)$ with gauss curvature zero, an isometric immersion $v:M\rightarrow \mathbb{R}^3$ that is $C^{1,\alpha}$ and I consider the Gauss ...
4
votes
1answer
207 views

Are “Unions” of small exotic $\mathbb{R}^4$'s small?

Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$. Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup ...
2
votes
1answer
111 views

Intersection of two real polynomial surfaces

Consider two real polynomials in three variables, defined on the 3-sphere, $S^3$. Is there some Bezout-type theorem, relating the intersection of two closed surfaces defined by these polynomials and ...
0
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0answers
43 views

Disturbing regular level submanifold of a smooth function

Let $a$ be a regular value of a smooth function on a closed manifold and $\{f=a\}$ a corresponding level submanifold. It is known that any such function can be approximated by a Morse function $g$. ...
1
vote
1answer
129 views

Generalized spin connection and dreibein in higher spin gravity

I am studying higher spin gravity and I would like to know the mathematical and physical meaning of generalized spin connection and generalized dreibein that appear in this theory. It is well known ...
1
vote
2answers
155 views

Local structure of the quotient of a Lie group action

Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure. Are there any results for the general case? (a) If the ...
1
vote
0answers
74 views

Determine the shape of curve which has the minimum number of inflection points

Let $A$ be a set of generic $C^2$ closed curves in $2$-dimentional Euclidean space, and an equivalence relation between $2$ curves $a$ and $b$ in $A$ is defind that there exist diffeomorpfic map $h ...
0
votes
1answer
86 views

Set of critical values is compact [closed]

Let $M\subseteq \mathbb{R}^n$ be a compact manifold with $\partial M=\emptyset$ and let $f: M\rightarrow S^p$ be a smooth map. I want to unerstand the proof of the following lemma which occurs in ...
4
votes
1answer
350 views

Cap product à la Poincaré

Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found ...
4
votes
1answer
307 views

Is the space of real conics with a singular point an orientable manifold?

Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set of such a polynomial gives a real curve ...
0
votes
1answer
115 views

Surjectivity of “nice maps” from local properties

What tools are available from real algebraic geometry, analysis and topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$ from local properties and maybe function values? ...
3
votes
1answer
388 views

What is wrong with the “naive” proof of the Hauptvermutung?

The Hauptvermutung is the statement that any two PL structures on a topological space have a common refinement. It is false in general, but (I think) true for some low dimensional manifolds. The ...
2
votes
1answer
166 views

Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies Normality ...
2
votes
0answers
161 views

Example of symplectic 4-manifolds with no Lefschetz fibration structure?

I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration ...
8
votes
2answers
441 views

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated?

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of ...
19
votes
0answers
259 views

Can one properly embed a differential manifold into numerical space of double dimension? [duplicate]

If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$. This is a not too difficult theorem due to Whitney, proved in many textbooks. ...
-1
votes
1answer
197 views

Are compact complete geodesics closed? [closed]

note: I find this question In stackexchange math, I would be interest to know how I could be answer this kind of question,I pasted it here as I see it appropriate For MO. check this link: ...