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”.

learn more… | top users | synonyms

4
votes
1answer
282 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
279 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
100 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
0answers
270 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
147 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
125 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 ...
-1
votes
0answers
26 views

If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that whenever $s(p,q) =0$ then $$ \nabla ...
8
votes
2answers
415 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
250 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
191 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: ...
1
vote
1answer
141 views

Normal Variation on Manifolds

Let $M$ be a smooth surface and let $x,y \in M$. Let $d_{M}(\cdot, \cdot)$ be the geodesic distance metric on $M$, that is the length of the shortest geodesic curve on $M$. Let $\kappa$ be the maximum ...
4
votes
0answers
233 views

If 2-manifolds are homeomorphic and smooth, are they diffeomorphic? [closed]

Perhaps this question has already been asked on Mathoverflow. I mean this question in a global sense. A friend mentioned it to me today, and I started thinking about it. I'm not sure how to prove it. ...
4
votes
1answer
53 views

Existence of an equivariant Morse function

Let $G$ be a (finite) group and $M$ a $G$ manifold. Now I have a smooth real valued function $f: M\rightarrow R$ with $f(x)=f(g(x)),\, \forall g\in G$. Now in general $f$ will maybe not be a Morse ...
1
vote
1answer
129 views

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that ...
5
votes
0answers
187 views

Gompf's invariant of $2$-plane fields

I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...
2
votes
1answer
61 views

Constructivity of zeros demanded by topological degree

Let $f : S^{n - 1} \to S^{n - 1}$ be a smooth map from the unit vectors of $\mathbb{R}^n$ to themselves. If $f$ has nonzero degree, then we know that any smooth map $g : D^n \to \mathbb{R}^n$ ...
0
votes
0answers
84 views

Degree of Map between Pseudomanifold

There are two different ways to define a degree of map. Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
3
votes
0answers
90 views

When does a leaf space admit a (non-Hausdorff) manifold structure?

If $f:M \to N$ is a submersion with connected fibers, then the fibers of $f$ foliate $M$. This is called a simple foliation of $M$ and the leaf space can be identified with $N$. Suppose that a ...
1
vote
1answer
124 views

Topology of surfaces and mean curvature

The Gauss-Bonnet theorem characterizes topology of surfaces by their Gaussian curvature. Do there exist results characterizing topology of surfaces embedded in $\mathbb{R}^3$ by their mean ...
0
votes
0answers
41 views

Characterization of global sections (which are not products) of a sheaf which is locally a product

In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
4
votes
2answers
123 views

functions which covers(good covers) manifolds

Let $M$ be a (not necessarily compact)) smooth manifold. 1.Is there a smooth map $f:M\to \mathbb{R}$ and an open covering $\mathbb{R}=\cup U_{\alpha}$ such that each $f^{-1}(U_{\alpha})$ ...
27
votes
0answers
484 views

Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$: $n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...
6
votes
0answers
241 views

Classifying space of the higher-structure diffeomorphism group

There is a higher extension of the classifying space $B \mathrm{Diff}$ of the diffeomorphism group implicit in the (infinity,n)-category of cobordisms with (X,zeta)-structure ...
6
votes
2answers
417 views

Reference for a fact (?) on homeomorphic knot complements

Does somebody have a reference (or an argument why it should be true) for the following statement? “Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 ...
11
votes
1answer
257 views

Diffeomorphisms and homotopy equivalences sliced over BO(n)

There are some classical results stating sufficient conditions on a manifold $\Sigma$ such that every homotopy equivalence $\Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)$ is homotopic to ...
0
votes
0answers
114 views

Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
0
votes
1answer
83 views

About the regularity of the boundary of a set

Consider $\Omega_1, \Omega_2$ two convex, bounded domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ and suppose that $\operatorname{dist}(x, \partial \Omega_2) =\min_{y \in \partial ...
14
votes
5answers
547 views

Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?

First, we make the following observation: let $X: M \rightarrow TM $ be a vector field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e. $$ i_X \circ i_{X} ...
1
vote
1answer
78 views

Topological description of the regular values of a differentiable function

Is there some kind of description of the set of regular values of a differentiable function $f:\mathbb{R}^{n} \to \mathbb{R}^{m}$ in topological terms? In particular, is the set of regular values ...
5
votes
1answer
110 views

Ehresmann's fibration theorem in the C1 class

I have seen on the French Wikipedia that Ehresmann's fibration theorem is stated with the assumption that everything is $C^2$, see Théorème de Ehresmann. (On the English Wikipedia, the assumption is ...
7
votes
0answers
112 views

Topological restrictions from mean curvature bounds

Alexandrov's Theorem says that a compact constant mean curvature hypersurface embedded in $\mathbb{R}^{n+1}$ must be a round sphere. What happens when the mean curvature is small, or bounded? (For ...
1
vote
0answers
78 views

Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here: ...
8
votes
2answers
534 views

Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?

The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...
1
vote
1answer
96 views

Classifying tensor powers of tangent bundles as pullbacks of appropriate tangent bundles

Assume we are given a smooth manifold $M$ and let $TM^{\otimes r}, r>1,$ be some tensor power of its tangent bundle. Is there any general observation/result saying when there exists a manifold ...
13
votes
1answer
336 views

Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...
9
votes
0answers
108 views

Detecting invertible elements in group rings by their images for finite quotients of the group

Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent). Consider an element ...
6
votes
0answers
79 views

Can one detect smoothness of $k$-forms with $k$-dimensional manifolds

Fix an integer $k\ge 1$. Let $X$ be a smooth manifold. It is well-known, that a real valued function on $X$ is smooth if its restriction along all smooth maps $M\to X$ for manifolds of dimension $\le ...
3
votes
1answer
80 views

cartesian product rigidity for the punctured open disc

Q1: Let $D^n$ ($n\geq 1$) be the n-dimensional open disk. If $D^n-\{0\}$ is homeomorphic to $X\times (0,1)$, for some topological space $X$, does it necessarily follow that $X$ is homeomorphic to ...
6
votes
1answer
237 views

Immersion of $S^1$ in $\mathbb{R}^2$ that can be extended to $\mathbb{D}$

I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow ...
1
vote
0answers
72 views

Decomposition of Lefschetz fibrations into surface bundles and Lefschetz fibrations over spheres

Assume that $M$ is a Lefschetz fibration with fiber genus $g$ over a base of genus $h>0$ (with at least one singular fiber). What obstructions exist to decomposing $M$ into a surface bundle $S$ of ...
1
vote
0answers
96 views

Simply-connected 4-manifolds can be blown up and down to complex projective planes. How about non-simply-connected ones?

There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$: $$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# ...
1
vote
0answers
62 views

Branch point and alexandrov embeddedness

This is a question I have asked on mathstackexchange with a bounty but without any answer; it is probably more adapted to mathoverflow: Let us assume that $\Sigma_n$ is a sequence of topological ...
3
votes
1answer
161 views

Self-diffeomorphisms of $\mathbb{R}^2$

I guess my question is very vague. Let $\phi: \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation-preserving diffeomorphism that fixes the origin. Does $\phi$ take any sort of "normal form" if we allow ...
4
votes
1answer
92 views

Differential topology, maximal isotropy of a manifold

I am interested in the degree of isotropy of a connected (by arc) manifold in general. Is it true that every connected manifold M (of dimension n) is maximally isotropic in the sense that you can ...
3
votes
0answers
86 views

Brouwer fixed points via flow

Let $B$ be a closed ball in $n$-dimensional space, let $f$ be a "sufficiently smooth" map from $B$ to $B$, and let $p$ be a point on the boundary of $B$. It seems to me that something like the ...
5
votes
2answers
245 views

What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...
3
votes
0answers
210 views

A lifting problem

Let $E\overset{\pi'}{\longrightarrow} B'$ and $E\overset{\pi}{\longrightarrow} B$ be vector bundles. For $i=0,1$, let $f_i$ be a fiber-preserving open embeddings of $\pi'$ into $\pi$, with $g_i$ the ...
0
votes
0answers
61 views

Geometric Proof of the Hirzebruch Signature Theorem (i.e. using Connections)

Is there a geometric proof of the Hirzebruch Signature Theorem, that is to say: Is the a proof that does not use the topological definition of the Chern/Pontrijagin classes, but only the Chern--Weil ...
1
vote
0answers
86 views

Thom-Pontryagin construction for pairs

Is there a generalization of the Thom-Pontryagin construction in the following sense? Let $M$ be a smooth manifold, $\partial M=A\cup B$ where $A$ and $B$ are $m-1$-manifolds with a common boundary ...
12
votes
0answers
173 views

What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times ...