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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6
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1answer
40 views

Is there any result on the “counting” of minimal atlas?

Take a differentiable manifold $M$. Define $\eta(M)$ as $\min\{\#\mathfrak{A} \mid \mathfrak{A} \text{ is an atlas for $M$}\}$. For example, if $M=S^n$, we have that $\eta(M)=2$, since $S^n$ is ...
1
vote
0answers
15 views

singular $1$ cube - Boundary of $2$ chain

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
0
votes
0answers
23 views

Independence of parametrization in defining integral of differential form

This is an exercise from Spivak's Calculus on Manifolds. Questions asks the following : (Independence of parametrization). Let $c$ be a singular $k$ cube and $p:[0,1]^k\rightarrow [0,1]^k$ be ...
5
votes
1answer
33 views

Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class ...
0
votes
0answers
63 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
2
votes
0answers
29 views

Poincaré duality isomorphism maps cohomology to homology here?

Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i: M \to A$. Let $k = p - n$. Does the Poincaré duality isomorphism$$\bigcap \mu_A: H^k(A) \to H_n(A)$$map the ...
2
votes
1answer
54 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...
11
votes
3answers
718 views

Example of a category without product

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
1
vote
0answers
27 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
3
votes
0answers
25 views

Let $M$ be a manifold noncompact. Then there is a closed embebdding of the half line $[0, \infty)$ into $M$.

Let $M$ be a manifold connected hausdorf noncompact. Then there is a closed embebdding of the half line $[0, \infty)$ into $M$. I'm hard to build such a function without self-intersections. Book: ...
1
vote
1answer
36 views

Is the associated bundle construction a bifunctor?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and ...
2
votes
0answers
46 views

Cohomology algebra generated by Steifel-Whitney classes and dual classes subject to defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
3
votes
0answers
38 views

Correspondence defines embedding of $G_n(\mathbb{R}^m)$ into $G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = G_{n+1}(\mathbb{R}^{m+1})$? [closed]

Does the correspondence$$X \overset{f}{\to} \mathbb{R}^1 \oplus X$$defines an embedding of the Grassmann manifold $G_n(\mathbb{R}^m)$ into $$G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = ...
1
vote
0answers
24 views

Does a map with fibers $S^2\vee S^2$ have to be a locally trivial $S^2\vee S^2$ bundle?

Let $X\to Y$ be a proper map between pseudo-manifolds such that fibers are $S^2\vee S^2$, is it true that $X\to Y$ is locally trivial $S^2\vee S^2$ bundle?
0
votes
1answer
41 views

Can we get a torus by identifying surface with removed disc and mobius strip?

If we take a surface and remove a disc, then identify this resulting circle with the boundary circle.. does this produce a torus?
5
votes
3answers
187 views

Can a Compact Lie Group have a Non-Compact Lie Subgroup?

Hopefully the title says it all in terms of the question I'm asking. I feel that the answer is no, but I'm not sure why; I don't really know many deep theorems about the structure of lie groups.
4
votes
1answer
69 views

Complement of a knot that *isn't* rationally null-homologous

Let $K$ be a knot in a closed, oriented 3-manifold $Y$. It is a standard fact that if $K$ is (at least rationally) null-homologous, then $H_1(Y-K;\mathbb{Z})$ is isomorphic to $H_1(Y;\mathbb{Z})\oplus ...
1
vote
1answer
61 views

Self-indexing Morse function on a torus which is a height function

A Morse function $f: \Bbb T^2\to [0,2]$ is called self-indexing if $f^{-1}(n)$ is the set of critical points of index $n$. It is relatively easy to see that on any compact manifold, any Morse function ...
0
votes
2answers
19 views

Brouwer degree extension Lemma

Let $M$ and $N$ be oriented $n$-dimensional manifolds without boundary an also $M$ is compact and $N$ connected. Suppose that $M$ is the boundary of a compact oriented manifold $X$ and that $M$ is ...
2
votes
0answers
32 views

Parameterization and geodesics of a 3-torus

So I'm thinking about a space exploration game where the primary mechanic is to fly a space ship around the surface of various 4-dimensional surfaces. The way I'd like to render this is by ray casting ...
3
votes
0answers
34 views

$\mathbb{R}^n$ bundle over $B$ being compact and having finite covering dimension implies has finite type? [closed]

Let $\xi$ be a $\mathbb{R}^n$-bundle over $B$. If $B$ is paracompact and has finite covering dimension, does it follow that every $\xi$ over $B$ has finite type?
1
vote
0answers
36 views

Three complex and Euler characteristic zero

So this is an excerpt from Thurston's three manifolds text. He goes onto state that by constructing a complex by gluing faces of polyhedra we have the following condition. Such a complex is a manifold ...
5
votes
1answer
36 views

Can any smooth function be written in this form?

Can any smooth function $F: \mathbb{R}^n \to \mathbb{R}$ be written in the form$$F(x) = F(a) + \sum_{\mu = 1}^n (x^\mu - a^\mu)H_\mu(x),$$where $a = (a^1, \dots, a^n) \in \mathbb{R}^n$ and the $H_\mu$ ...
-2
votes
1answer
38 views

A function $\phi$ between two manifolds of class $C^\infty$ is constant if $d\phi$ [closed]

Let $M$ and $N$ two manifolds of $C^{\infty}$, $M$ connected, and $\phi:M\to N$ also of class $C^{\infty}$ so that in all point $m\in M$ the function $d\phi(m):M_m\to N_{\phi(m)}$ between the ...
0
votes
0answers
40 views

Brouwer Degree is locally constant

I'm reading Milnor's book "Topology from the differential viewpoint" and I'm stuck at this point: Let $M$ and $N$ be oriented n-dimensional manifolds without boundary and let $$f: M \longrightarrow ...
4
votes
2answers
62 views

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$?

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$? For starters, I know that if the $n$-dimensional $M$ can be immersed in ...
1
vote
0answers
45 views

normal bundles are stably isotopic

I am searching for a reference about the following theorem: Let $M \xrightarrow{i_{0}} \mathbb{R}^{k},\,\,$ $M \xrightarrow{i_{1}} \mathbb{R}^{l}$ be two embeddings of the manifold $M$. We know by ...
0
votes
0answers
18 views

Non contable of homeomorphism family of the unitary open ball in itself

Let $M$ a variety of dimension $n$. Show that if have a structure of class $C^{\infty}$ then have a not countable number of such structures. entonces posee una cantidad no-numerable de tales ...
0
votes
1answer
49 views

Guillmin & Pollack's Definition of a Manifold

In Guillemin and Pollack's Differential Topology, they (roughly speaking) define a manifold to be a space which is locally diffeomorphic to Euclidean space. Now this is obviously not the full ...
5
votes
1answer
66 views

Lie groups where $x \mapsto x^2$ is a diffeomorphism?

In every Lie group $G$ the function $x \mapsto x^2$ is a local diffeomorphism in a neighbourhood of the identiy. (This is because its differential is: $v \mapsto 2v$ when considered as a map from ...
1
vote
0answers
36 views

Monomorphism from a sheaf to a flasque sheaf: determining the stalk.

Let $M$ be a topological Hausdorff space. We use the following definitions (as they may vary): A presheaf $\mathcal{F}$ is a collection of vector spaces $\mathcal{F}(U)$ for each open subset $U$ of ...
2
votes
1answer
62 views

Injective immersion that is not trajectory of any flow

Let $M$ be a compact manifold of dimension $m \geq 2$. Show that there exists an injective immersion of $\mathbb{R}$ in $M$, whose image is not the trajectory of any flow. I know how to do it for ...
0
votes
0answers
25 views

Submersion and some properties

Theorem: Let $f:M\to\mathbb R^m, M\subset\mathbb R^n$ be a submersion, $p\in M$ and $D_pf:\mathbb R^n\to\mathbb R^m$ is the functionalmatrix. Then there exists: - an open neighborhood $A$ on ...
0
votes
0answers
25 views

Comparison of orientations involving diagonals

This problem came up in a discussion about orientations and and seems more delicate than I expected: Let $M_1$, $M_2$ and $P$ be smooth oriented finite-dimensional manifolds without boundary. Let ...
5
votes
1answer
47 views

Components of the space of immersions 2-manifold into $\mathbb R^3$

Let $M$ be a $2$-sphere with $g$ handles. Consider the space of maps $M\to \mathbb R^3$, which are immersions [i.e. smooth maps with nondegenerate differential in each point $x\in M$], with ...
1
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0answers
30 views

convergence in the $C^r$-topology on $C^r(M,N)$ for $M$, $N$ compact manifolds

Let $M$ and $N$ be smooth (finite dimensional) manifolds without boundary. On the set $C^r(M,N)$ we choose the compact-open $C^r$-topology. This topology is defined as follows (I take the definition ...
6
votes
3answers
181 views

For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
2
votes
1answer
122 views

Why do these two facts imply that $S^n$ is not parallelizable for $n$ even, $n \ge 2$?

Consider the following two facts. If $S^n$ admits a vector field which is nowhere zero, the identity map of $S^n$ is homotopic to the antipodal map. For $n$ even, the antipodal map of $S^n$ is ...
8
votes
1answer
116 views

Is a general smooth rescaling of a complete vector field itself complete?

$\newcommand{\Ga}{\Gamma}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\til}{\tilde}$ $\newcommand{\M}{M}$ $\newcommand{\ep}{\epsilon}$ $\newcommand{\brk}[1]{\left(#1\right)} $ ...
3
votes
2answers
79 views

$S^n$ admitting nowhere zero vector field implies identity map of $S^n$ is homotopic to antipodal map? [closed]

If $S^n$ admits a vector field which is nowhere zero, does it follow that the identity map of $S^n$ is homotopic to the antipodal map?
3
votes
1answer
42 views

Is parallelizability equivalent to the set of vector fields being free?

We have the $C^{\infty}(M)$-module $\mathcal{D}^1(M)$ of vector fields over a $C^{\infty}$ manifold $M$. Is being parallelizable equivalent to this module being free, of dimension $n$? I have the ...
2
votes
1answer
46 views

Are the two standard descriptions of $\mathbb{C}P^{\infty}$ (topologically) equivalent?

While reading through some issues of Baez's (wonderful) "This Week's Finds in Mathematical Physics," I came across this statement (from week 149): $K(\mathbb{Z},2)$ is a bit more complicated: it's ...
1
vote
1answer
33 views

When are two vector fields $C^1$ close?

For simplicity let us assume we have a fixed compact manifold M. Introduction: While considering $C^\infty(M,\mathbb{R})$ one can say that two functions $f,g$ are $\epsilon-close$ iff for fixed ...
0
votes
0answers
17 views

Parallely transported vector continuous with respect to homotopy of curve?

Let $M$ be a smooth manifold with connection. Choose two points $p,q \in M$ and connect them with curve $\gamma _0$. Suppose I take fixed vector at $x$ and parallel transport it to $y$. Denote this ...
3
votes
1answer
43 views

Inward/Outward-pointing tangent vector is well-defined

Let $M$ be a smooth manifold with boundary and $p\in \partial M$. We say a tangent vector $v\in T_pM$ is inward-pointing if in a chart $x$ with $v=v^i\partial/\partial x^i$ (using the summation ...
26
votes
1answer
341 views

“the only odd dimensional spheres with a unique smooth structure are $S^1$, $S^3$, $S^5$, $S^{61}$”

This (long) paper, Guozhen Wang, Zhouli Xu. "On the uniqueness of the smooth structure of the 61-sphere." arXiv:1601.02184 [math.AT]. proves that the only odd dimensional spheres with a ...
12
votes
3answers
272 views

What's true in $\mathbb{R}^4$, false in $\mathbb{R}^3$ and uninteresting in $\mathbb{R}^5$?

What are some interesting and easy-to-understand (for non-differential geometers) facts about subobjects of $\mathbb{R}^4$ that are not only false in $\mathbb{R}^3$, but also specific to the structure ...
1
vote
1answer
36 views

What does the notation $g\cdot\omega$ mean in Spivak's Calculus on manifolds?

In chapter $4$ (Integration on chains) of Spivak's Calculus on manifolds he says the following: If $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is differentiable and $\omega$ be a $k$ form on ...
5
votes
1answer
65 views

Smooth manifold which is a group, but not a Lie Group

Are there (preferably non-pathological) examples of smooth manifolds, which are groups, but not Lie groups? In books one can see plenty of examples of Lie groups, but I haven't seen an example where ...
2
votes
0answers
57 views

When are differential forms related by a base space automorphism?

Let $w$ and $u$ be nowhere-vanishing smooth differential forms fields of degree $n$ on a smooth manifold $M$ (aka smooth sections of $\Omega^n(M)$). When does there exist an automorphism $f: M \to M$ ...