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.

learn more… | top users | synonyms

1
vote
2answers
72 views

Regular points on sphere.

I am reading Milnor's book "Topology from the Differentiable Viewpoint" and have a question of regular values. So, he defines a regular point as follows. Let $f :M \to N $ be a smooth map between ...
0
votes
0answers
22 views

Alternative Definition of Cotangent Space [duplicate]

Let $M$ be a smooth manifold, $p \in M$, and let $I_p$ be the set of all smooth functions $f: M \rightarrow \mathbb{R}$ such that $f(p)=0$. Define $I_p^{2} = \{\sum_{i=1}^{k}f_ig_i: f_i, g_i \in ...
1
vote
0answers
31 views

Linearisation in direction of formal adjoint

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 ...
0
votes
1answer
35 views

Question about definition of vector field

The following definition of vector fields were given in class, A $\textbf{Smooth Vector Field on a manifold $M$}$ is a smooth function, $$s: M \to TM$$ such that $$\pi \circ s=id_M$$ We then went ...
2
votes
1answer
38 views

Construction of a smooth function

I know this question is related to partitions of unity, but we never actually discussed these in class. Suppose $M$ is an abstract smooth manifold, $K \subset U \subset M$ an inclusion of a compact ...
1
vote
0answers
79 views

Proof of a lemma is unclear to me (Theorem of inverse functions)

Lemma: Let $B(a,r)$ be a ball in Banach space $X$ and $\phi$ be a contraction ($d(\phi(x),\phi(y)\leq qd(x,y),0<q<1$) from $B(a,r)\to X $. Then the function:$$ f(x)=x+\phi(x)$$ is a ...
1
vote
1answer
35 views

Reference for transversality — relative version

In Hirsch, Differential topology, Thm 2.1 states that for smooth manifolds $M,N$ and a closed submanifold $A\subseteq N$, the space of maps from $M$ to $N$ transverse to $A$ is dense and open in ...
1
vote
0answers
32 views

Local bisections of Lie groupoids

Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...
0
votes
0answers
7 views

$f\in C^{\infty}(0,\infty)$ and $f^j(x)=O(x^-a_j)$ with $0\le a_j\lt1$

If $$f\in C^{\infty}(0,\infty)$$ and $$f^j(x)=O(x^-a_j)$$ with $$0\le a_j\lt1$$ then prove that $$f\in C^{\infty}[0,\infty)$$ I do not know how to start to prove this any ideas?
0
votes
0answers
25 views

If $f\colon X\to Y$ is nullhomotopic, then $I_2(f,Z)=0$ for closed $Z\subset Y$ of complementary dimension?

I think I have a near solution, but one doubt. Problem 2.4.4 on page 83 of Guillemin and Pollack asks If $f\colon X\to Y$ is homotopic to a constant map, show that $I_2(f,Z)=0$ for all ...
2
votes
0answers
84 views

A(nother ignorant) question on phantom maps

My last question (Is such a map always null-homotopic?) is quite similar. If you do not care about my motivation for these questions, you can skip to the last line. I asked if some assumptions were ...
3
votes
1answer
27 views

Framing of embedding induces an isotopy of embeddings

Let $M$ be a smooth manifold of dimension $m$ and $\phi : S \to M$ a smooth embedding (dim S = k < m) such that the normal bundle $T_S M$ is trivializable. Let $f: T_S M \to S \times ...
1
vote
1answer
41 views

Pullbacks of smooth densities through smooth functions are smooth?

Problem Statement This is an in-text exercise from John Lee's Smooth Manifolds book chapter 14. If $F: M \rightarrow N$ is a smooth function from manifolds $M$ to $N$, and $\mu$ is a smooth density ...
1
vote
1answer
50 views

What is the derivative of the map $\mathbb{R}^\ell\times S\to\mathbb{R}^N$ given by $(t_1,\dots,t_\ell,v_1,\dots,v_\ell)\mapsto \sum t_iv_i$?

In problem 7, pg. 75 of Guillemin and Pollack's Differential topology, there is a hint saying The set $S\subset(\mathbb{R}^N)^\ell$ of linearly independent $\ell$-tuples in $\mathbb{R}^N$ is open ...
7
votes
2answers
75 views

What's the “easiest” closed 3-manifold with a nonabelian fundamental group?

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group. It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one ...
2
votes
1answer
58 views

triviality of tensor product of vector bundles

Let $\xi$ be a $O(n)$-bundle with fibre $\mathbb{R}^n$. Let $\xi\otimes \mathbb{C}$, $\xi\otimes \mathbb{H}$ be complex vector bundles and quaternionic vector bundles. If $\xi$ is not a trivial ...
0
votes
0answers
27 views

principal bundle and the associated bundle

Let $G\leq O(n)$ be a subgroup of orthogonal group. Let $\xi$ be a principal $G$-bundle. Let $\xi[\mathbb{R}^n]$ be the associated vector bundle. If $\xi$ is not a trivial bundle, can we obtain that ...
3
votes
1answer
47 views

Question on tangent spaces

In this question, if I also had that $f$ were a diffeomorphism and $f^k = I$ for some positive integer $k$ would it make a difference to the answer being in the negative? Here, by $f^k$ I mean $f$ ...
1
vote
1answer
44 views

Polynomials of a fixed degree have a nonzero partial at all points of their vanishing set?

I'm having difficulty with the second half of a question from an old homework assignment (for a differential geometry class I am currently taking). The first half of the question asked me to assume ...
0
votes
0answers
49 views

Coordinates at a singularity

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a smooth function. Let's assume that $f$ has a local minimum at $p \in \mathbb{R}^2$ and hence $| \nabla f|\ = 0$ at $p.$ Intuitively, one should be able to ...
2
votes
1answer
42 views

The identity map on a tangent space

If $X$ is a smooth manifold and $I : X \rightarrow X$ is the identity map on $X$ (with the same smooth structure on both sides) then I can show that $dI_x : T_x X \rightarrow T_x X$ is the identity ...
1
vote
0answers
38 views

Subdifferential of a convex function

How would I find a convex function $f: \mathbb{R} \to \mathbb{R}$ such that $\partial f(0) = [0,1]$ A subdifferential is just the collection of vectors $w \in \mathbb{R}^n$ such that $f(y) \geq ...
0
votes
1answer
69 views

Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions:

Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions: a. There is another point $p_{0}$ such that the ...
0
votes
2answers
55 views

Immersion, embedding and category theory

The map between smooth manifolds $f:M \to N$ is called an immersion if the tangent map $Tf$ is injective for each $x \in M$. There is a corresponding notion of submersion. About embeddings I met two ...
0
votes
1answer
66 views

Immersion $\mathbb{S}^n\times\mathbb{R}\to\mathbb{R}^{n+1}$

Immersion $\mathbb{S}^2\times\mathbb{R}\to\mathbb{R}^3$ As $\mathbb{S}^2\times\mathbb{R}$ it's not compact, i can give immersion given by $$(x,y,z,t)\to e^t(x, y, z).$$ or i'm wrong. Could you give ...
0
votes
1answer
36 views

Interpretation of local submersion theorem

Wikipedia gives the following formulation of the $\textbf{Local Submersion thoerem}$, If $f: M \to N$ is a submersion at $p$ and $f(p)=q \in N$, then there exists an open neighborhood $U$ of $p$ in ...
1
vote
1answer
41 views

Embedding the mapping $\phi:\mathbb{CP}^n\times{\mathbb{CP}^m} \to\mathbb{CP}^{nm+n+m}$

The canonical coordinates of $\mathbb{CP}^n$ are $x=[(x_0,\ldots, x_n)]$, $x_i\in\mathbb{C}$. How to prove that the mapping $$\phi:\mathbb{CP}^n\times{\mathbb{CP}^m} \to\mathbb{CP}^{nm+n+m}$$ ...
1
vote
1answer
52 views

Manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$.

Let $H=\{(x, y, z)\in\mathbb{R}^3 | \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^2}=1\}$. Prove that $H$ is a manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$. Already I tried to be spread ...
2
votes
1answer
43 views

ways to see whether the Pontryagin class of a quaternionic line bundle over a CW-complex is zero

the first pontryagin class of a quaternionic line bundle over a CW-complex is zero if and only if the quaternionic line bundle is trivial or not? Let $\xi^\mathbb{H}$ be a given quaternionic line ...
1
vote
0answers
79 views

Question about Milnor's proof of Sard's Theorem

We've just covered Sard's theorem and have just started to look at transversality in my differential geometry class and I'm trying to understand a proof of Sard's theorem (based on Milnor's proof): If ...
1
vote
0answers
32 views

Open cover of manifold with boundary

If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold $M$, then each $U_{\alpha}$ is a smooth manifold. I want to extend this fact to manifolds without boundary. So my ...
4
votes
2answers
168 views

Winding number integral/index in plane

Let $B(x_0,y_0)$ be an open unit disk. Assume that $F(x,y)=(f_1 (x,y),f_2 (x,y)):\overline{B(x_0,y_0)}\rightarrow \mathbb{R}^2$ is a diffeomorphism. Assume also that $F(x,y) \ne (s_0,t_0) $, for all ...
3
votes
0answers
63 views

When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
1
vote
1answer
38 views

Mapping degree of a diffeomorphism

This might be a bit silly question but I haven't find direct reference. Let $\Omega$ be open, bounded and connected in $\mathbb{R}^n$. Assume that $f:\overline{\Omega}\rightarrow \mathbb{R}^n$ is a ...
2
votes
0answers
49 views

Smooth manifolds having an analytic structure.

I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are ...
4
votes
2answers
114 views

chern class of complex line bundle

Let $\xi:=(\mathbb{C},E,p,B)$ be a complex line bundle, where $B$ is a manifold or CW-complex. How to determine whether the first Chern class $c_1(\xi)=0$ or non-vanishing?
1
vote
1answer
51 views

Lemma on locally finite open covers

I came across this lemma in Lee's 'Introduction to Smooth Manifolds'. The lemma seems simple enough to prove, but I just can't seem to prove it. It's frustrating me because I know it must be simple. ...
0
votes
0answers
38 views

integral cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
5
votes
1answer
48 views

Theorem 3.1 from Milnor's Morse Theory

Milnor is in the business of proving that if $f: M \to \mathbb{R}$ is a smooth function, $a < b$, and $f^{-1} ([a,b])$ is a compact subset of $M$ containing no critical points, then $M^a$ is ...
2
votes
1answer
64 views

Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
2
votes
1answer
63 views

Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
1
vote
0answers
58 views

Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
0
votes
2answers
94 views

Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
4
votes
2answers
140 views

Invertibility theorem on the boundary for a function between two closed 2D manifolds

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
4
votes
2answers
100 views

What is the anticommutator of the interior product and codifferential (adjoint of exterior derivative)?

What is $\eta=i_X \delta + \delta i_X$ acting on differential forms? Here $\delta$ is the usual Hodge adjoint of the exterior derivative and $i_X$ is contraction of a form with the vector field $X$. ...
4
votes
1answer
53 views

Map between Tangent Manifolds Well-Defined?

Let $f: \mathcal{M} \to \mathcal{N}$ be a $\mathscr{C}^{r+1}$ map. We define a map $\mathscr{T}f: \mathscr{T}\mathcal{M} \to \mathscr{T}\mathcal{N}$ as follows: A local representation of the map ...
0
votes
3answers
77 views

Why does the slope of a smooth simple closed curve have winding number one?

$\def\RR{\mathbb{R}}$Let $S^1$ be the circle and let $\gamma : S^1 \to \RR^2$ be a smooth injective map with $\gamma'(t)$ everywhere nonzero. What is the easiest way to show that $t \mapsto ...
0
votes
0answers
53 views

Ambient isotopy of based surface knots

Let $S$ be a smooth closed surface of genus $\ell$. Let $p$ be a point of $S$ and $a_i$, $b_i$ with $i=1,\ldots,\ell$ be $2\ell$ curves embedded in $S$ based at $p$ smooth everywhere except perhaps ...
1
vote
2answers
69 views

Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
4
votes
3answers
119 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...