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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4
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5answers
463 views

A Compact Hausdorff Space with no Manifold Structure? [on hold]

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
3
votes
1answer
60 views

Property of second Steifel-Whitney class?

Let $M$ be manifold, $n = 4$. Is $w_2$ special in in the regard it's the only thing of $H^2(M, \mathbb{Z}_2)$ where $w_2 \cup \tau = \tau \cup \tau$, $\tau \in H^2(M, \mathbb{Z}_2)$ or not? I wondered ...
0
votes
0answers
14 views

Partitions of unity subordinate to open cover of manifolds with boundary?

I am attempting to adapt Lee's proof of the fact that open covers of manifolds without boundary always admit smooth partitions of unity to the case in which the manifold does have boundary. The ...
0
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0answers
49 views
+50

Topology of Isolated Singularities

In Singular Points of Complex Hypersurfaces, Milnor shows that if $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated singularity at the origin, then $f^{-1}(\epsilon) \cap ...
0
votes
0answers
33 views

Topology for distributions on a compact space

I'm having trouble in distribution theory, though not in the usual setting. The context is in theoretical physics, trying to solve BF theory. My goal is to solve the following equation: $$\forall i ...
1
vote
0answers
29 views
+100

Linking circles inside an immersed surface

A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the unknot has the disk as a Seifert surface.) ...
8
votes
6answers
100 views

Examples of global properties that don't arise from local knowledge

Let $M$ be a smooth manifold. As an example of a global property that arises from local data we know that if $(M,g)$ is a compact surface without boundary then the Euler characteristic is given by $$ ...
2
votes
0answers
46 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
0
votes
1answer
11 views

Regarding embeddings and homological/cohomological injectivity

Possibly low-level question here: if one has an embedding $M\hookrightarrow X$, call it $\iota$, then this is of course an immersion, so the differential maps $\iota_{\ast}:T_{p}M\rightarrow ...
0
votes
1answer
68 views

Verify that this is not orientable.

Verify that this is not orientable. Möbius transformation: $$U=\{(t,\theta) \mid \frac{-1}{2}\lt t\lt \frac{1}{2}, 0\lt \theta \lt 2\pi \}$$ $\sigma (t, \theta)=<((1-t\sin (\theta/2))\cos ...
0
votes
0answers
19 views

Which of these plane curves are an immersion?

The question asks, which one of these plane curves is an immersion. I'm just checking that I'm correct. A is an immersion because the derivative is everywhere nonzero (thus the derivative is ...
3
votes
3answers
895 views

Holomorphic functions on a complex compact manifold are only constants

Is there a simple proof that every holomorphic function $M\to\mathbb{C}$ on a compact complex manifold $M$ is constant?
6
votes
1answer
46 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 ...
2
votes
0answers
27 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 ...
5
votes
1answer
53 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
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 ...
0
votes
0answers
66 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 ...
4
votes
1answer
289 views

Stiefel-Whitney numbers for product bundle

I'm reading Milnor's "characteristic classes" and I want to compute Stiefel-Whitney numbers of $ P^2 \times P^2 $ (product of projective spaces) for one of the problems, I know how Stiefel-Whitney ...
2
votes
0answers
42 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 ...
5
votes
3answers
94 views

Generalized Gauss map, giving rise to second fundamental form

I know that the tangent bundle of $G_n(\mathbb{R}^{n+k})$ is isomorphic to $\text{Hom}(\gamma^n(\mathbb{R}^{n+k}), \gamma^\perp)$, where $\gamma^\perp$ denotes the orthogonal complement of ...
2
votes
1answer
56 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 ...
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 ...
11
votes
3answers
730 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
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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: ...
5
votes
2answers
94 views

Smooth manifold $M$ is completely determined by the ring $F$.

For any smooth manifold $M$, the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and every point $x \in M$ determines a ring homomorphism $F ...
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 ...
6
votes
3answers
182 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. ...
8
votes
2answers
67 views

Diffeomorphism between $\mathbb{P}^n$ and the submanifold of $\mathbb{R}^{(n+1)^2}$ consisting of certain matrices?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinates space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$ by $q(x) = \mathbb{R}x ...
3
votes
2answers
46 views

Subset $V$ of projective space is open iff $q^{-1}(V)$ is open?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinate space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$by $q(x) = \mathbb{R}x =$ ...
8
votes
2answers
302 views

On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
2
votes
0answers
50 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
39 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?
4
votes
1answer
70 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 ...
0
votes
1answer
42 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.
1
vote
1answer
65 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
36 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?
4
votes
1answer
88 views

Grassmannian, symmetric, idempotent matrices of trace $n$?

How do I see that $G_n(\mathbb{R}^m)$ is diffeomorphic to the smooth manifold consisting of all $m \times m$ symmetric, idempotent matrices of trace $n$?
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
38 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
64 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 ...
6
votes
3answers
209 views

Fundamental groups of codimension 1 manifold complements

Let $M$ be a smooth manifold of dimension at most $3$ and $S \subset M$ a smoothly embedded compact connected codimension $1$ manifold, separating $M$ into two components, $M_1$ and $M_2$. I wonder ...
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 ...