Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Intuitive idea of immersions?

So I understand the definition of immersions and submersions, as well as the motivation for defining such ideas. Not only are they important in understanding properties of mappings of tangent spaces, ...
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129 views

why is there no non-degenerate 2-forms on 4-sphere?

The question is in the title. I have been told that there are actually no non-degenerate 2-forms on $S^{2n}$ for $n \neq 1,3$. I have found the following question: No symplectic structure on ...
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247 views

A question from Milnor's “Topology from a differentiable viewpoint”

Milnor's "Topology from a Differentiable Viewpoint" says the following: Let $f:M\to N$ be a smooth mapping, where $M$ is $m$ dimensional and $N$ is $n$ dimensional. Moreover, $m\geq n$. If $y\in ...
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44 views

The differential of a smooth map on manifold at points of local maxima

I have a differentiable function $f:M \to \mathbb{R}$ where $M$ is a smooth manifold. If $p \in M $ is a point of local maxima, that is I have an open set $V \subset M$, $p \in V$, so that $f(p)\geq ...
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46 views

Invariant forms on principal bundles

Let $\pi:M \to B$ be a principal $G$-bundle and $\xi$ a invarint $k$-form on $M$. Does $k> dimG$ implies that $\xi$ is a basic form (pull back of a $k$-form on the base manifold $B$)?
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Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
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48 views

Given points $x,y$, and tangent vectors $v,w$, is there a diffeomorphism $x\mapsto y$ and $v\mapsto w$?

Suppose $M$ is a smooth manifold, and you're given two points $x,y$, and associated tangent vectors $v$ and $w$. I'm curious, does there exist a diffeomorphism $F$ such that $F(x)=y$ and $dF_x(v)=w$? ...
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80 views

Representative of a cohomology class in once punctured solid torus

Consider a once punctured solid torus $(\mathbb R^2 \times S^1) /\{pt\}$. It is not difficult to see that it is homotopy equivalent to the bouquet of spheres $S^2\vee S^1$. So this guy has a ...
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59 views

Naturality of Lie derivatives

It was left to me as an exercise to show the naturality of the Lie derivative of arbitrary tensors on a smooth manifold. Is the following argument correct (it seems too easy)? Let $X$ be a vector ...
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1answer
93 views

Orientable surface bundles over the circle and their structure group

I want to understand whether orientable surface bundles over the circle, i.e. with orientable total space, are always trivial, so I though I would revive an old post and ask for a few clarifications, ...
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61 views

Degree of a vector field on the boundary of a punctured ball?

Suppose $v\colon \mathbb{R}^3\setminus\{x_1,\dots,x_n\}\to S^2$ is a smooth map, where all the points $x_i$ sit inside the unit ball $B^3$, and not on the boundary $S^2$. I suppose this map can be ...
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105 views

rudin's principles of mathematical analysis 10.31

I'm working on rudin's principles of mathematical analysis(3rd edition). There is problem too complex for me to solve.Please help me. The problem is on p270-271 of text. "Let T be 1-1 mapping of ...
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1answer
85 views

A doubt from Milnor's “Topology from a Differentiable Viewpoint”.

This is a doubt from Milnor's "Topology from a Differentiable Viewpoint". For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of ...
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69 views

Is this orientation preserving or reversing?

I am confused about the definition of orientation on manifolds. Let $X=\{(x,y,0)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ and $Y=\{(x,y,1)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ be two one dimensional circles in ...
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1answer
47 views

Trivial nature of orientable fibre bundle with cylinder base space

Given a bundle whose base space is a "cylinder", i.e. $\mathbb{S}^1 \times \mathbb{B}^n$ where $\mathbb{B}^n$ is an $n$-ball, is orientability (of the total space) enough to ensure that the bundle is ...
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1answer
56 views

Differential Structure of Two Dimensional Manifold

I am a beginner in Differential Topology, and have learnt that Theorem One dimensional differential manifold without boundary is diffeomorphic to $\mathbb R$ or $\mathbb S$. So I am curious ...
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159 views

Is there anything “nice” about the set of normal matrices (over $\Bbb R$ and $\Bbb C$?)

Normal matrices are of course useful to any linear algebra buff, not least because of the spectral theorem. However, taken as a whole, they tend to have some not-so-nice properties. For example: ...
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37 views

Orientability of space with only even dimensional cells.

I have the following question: Suppose a compact $\mathbb{R}$-manifold has finite cell decomposition with only even dimensional cells. Then $M$ is orientable. It's a theorem that a closed ...
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60 views

Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
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53 views

Simultaneous coordinate representation of a submanifold and its sub-submanifold

Suppose $Z\subset X\subset Y$ are manifolds and $z \in Z$. Prove that there exist an independent function $g_1,...g_l$ on a neighborhood $W$ of $z$ in $Y$ such that $$Z \cap W =\{y\in ...
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1answer
103 views

Prob. 10 of Chapter 8 of Milnor's Topology From a Differentiable Viewpoint. (Tangent Bundle is a Smooth Manifold.)

I want to solve Prob. 10 of Chapter 8 from Milnor's Topology From a Differentiable Viewpoint. The problem states that: Let $M\subseteq \mathbf R^k$ be a smooth manifold of dimension $m$. Show ...
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1answer
158 views

Does the cup product on de Rham cohomology induce a nondegenerate bilinear form?

I have small issue I came across in the following. Suppose $M$ is a compact, oriented manifold of dimension $4n+2$. I want to prove that the de Rham cohomology group $H^{2n+1}(M)$ are even ...
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100 views

Pullback of a form under the retraction $r\colon \mathbb{R}^n\setminus\{0\}\to S^{n-1}$.

The following is from Spivak's DG Lemma 7 in Chapter 8, but I'm muddled in a computation. Define two $(n-1)$-forms on $\mathbb{R}^n\setminus\{0\}$ by $$ ...
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49 views

How can we prove that $GL(2,\mathbb{R})$ is a topological group in $R^{4}$

A group is called topological group if it satisfies three properties 1) G is a Hausdorff space in K (Here we want to prove that if $ A,B \in GL(2,\mathbb{R})$ then we can find two disjoint open sets ...
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1answer
34 views

inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
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136 views

Questions about simplex and affine space

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
2
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1answer
73 views

Is the image under a homeomorphism of the cut locus $C_p$ a null-set?

Let $M$ be a complete Riemannian manifold with a point $p  \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of ...
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1answer
36 views

Taylor development on manifolds and Manifolds of differentiable Mappings?

I'm trying to study the book "Manifolds of Differentiable Mappings" written by Michor and I came across the following: He considers two smooth manifolds $M$ and $N$ and define an equivalence relation ...
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44 views

The space of collars of a manifold is contractible

Theorem: Let $M$ be a smooth manifold with boundary $\partial M$. Let $e_0,e_1 : \partial M\times [0,1]\rightarrow M$ be collars of $M$, i.e. $e_i$ are embeddings such that $e_i(x,0)=x$ for each ...
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How to get a Kirby diagram of $S^1 \times M^3$ if $M^3$ is given by a surgery diagram?

In "4-manifolds and Kirby Calculus" by Gompf and Stipsicz, there is a nice description of how to get the Kirby diagram of $S^1 \times M^3$, given a Kirby diagram of $M^3$. Basically, one thickens the ...
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1answer
106 views

Transversality, mod 2 degree, Winding numbers in differential topology

From Chapter 2 Section 5 of Guillemin and Pollack, Differential Topology, $\mathbf{X}$ is a compact connected manifold, and $f:\mathbf{X}\rightarrow \mathbb{R}^n$ a smooth map and $\dim{X}=n-1$. ...
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143 views

Minkowski space is locally Euclidean?

The Minkowski spacetime $\mathbb{R}^{1,3}$ is said to be a manifold (isomorphic to $SO^{1,3}$. But according to the definition of a manifold it should be locally euclidean. However, this seems to be ...
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1answer
528 views

Analytic approximations of the step function

Consider the Heaviside step function: $$H:\mathbb{R}\to \mathbb{R} $$ defined by $$H(x)=\begin{cases} 0 & \mbox{if } x<0 \\ 1 & \mbox{if } x\geq 0\end{cases}$$ Fix any $\delta>0$. Given ...
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$C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
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1answer
36 views

In uniform circular motion in R^2, is acceleration in the normal bundle?

In physics we learn that accleration is a vector quantity parallel to the radius and orthogonal to the velocity. With the embedding $\mathbb{S}^1 \hookrightarrow \mathbb{R}^2$ and the induced ...
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147 views

Homeomorphism of a Genus-2 Surface

Does there exist a homeomorphism from a genus-2 surface, the connected sum of 2 tori, to two circles, $S^1$, intersecting at a point? Intuitively it seems that the double torus can be squeezed into ...
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1answer
57 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
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3answers
109 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
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Why is there no natural metric on manifolds?

One of the things that always bothered me after learning introductory differential geometry (as a physics student) and then delving deeper into this field on my own is that, the usual construction of ...
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1answer
187 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
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How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
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124 views

Why does every noncompact orientable surface have a complex structure?

There is a high-powered proof of the fact that orientable noncompact surfaces have free fundamental group here that invokes the ability to put a complex structure on any such surface. But why should ...
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201 views

Are definitions for smooth map between manifolds are equivalent?

There are two ways to define a smooth map between manifolds. The 1st way (for example, Lee): $f:M\rightarrow N$ is smooth iff for every $p\in M$ there exist charts $(U,\varphi)$ at $p$ and $(V,\psi)$ ...
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How does a differential act when we identify $T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N$?

It's fairly common to identify the tangent space of a product manifold as $$ T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N $$ where $p=(p_1,p_2)$, and the actual isomorphism is given by $v\in ...
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If two Curves in $\Bbb R^3$ are transversal then they do not intersect

Proving this is easy: Let $X$ and $Z$ be the two curves in $\Bbb R^3$. Assume $X$ and $Z$ intersect at a point $y$. Then, at most $\dim(T_y(X) + T_y(Z)) = 2$, where $T_x(X)$ and $T_z(Z)$ denotes the ...
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1answer
151 views

If $S\subseteq M\times N$ is embedded, and $S$ and $\{p\}\times N$ intersect transversely in one point, then $\pi_M|_S$ is a diffeomorphism?

I'm trying to prove the equivalence of the following statements: Suppose $M^m$ and $N^n$ are smooth manifolds, $S\subseteq M\times N$ immersed, and $\pi_M$ and $\pi_N$ the projection maps. TFAE: ...
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109 views

Transition Functions for Cartesian Coordinate Systems

This is my first time using Mathematics SE (I've only used Physics and Astronomy before), so I apologize if this question is awkwardly phrased or incorrectly presented. I welcome any and all edits and ...
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2answers
71 views

Proving that a form is exact

Maybe this question is rather obvious but I didn't manage to solve it myself. Assume $M$ is a closed, oriented manifold. take $$ \Omega^k(M)\ni \omega = \begin{cases} d\beta~,~~~ in~ U\\ 0~,~~~ ...
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75 views

Does Heine-Borel hold for smooth manifolds?

If $M$ is a smooth $n$-manifold, the famous Whitney embedding theorems show that we can view $M$ as an embedded submanifold of some Euclidean space $\mathbb{R}^N$. Does the Heine-Borel theorem still ...
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117 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...