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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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 ...
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Topology of Isolated Singularities

In Singular Points of Complex Hypersurfaces, Milnor explains that if $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated singularity at the origin, then $f^{-1}(\epsilon) \cap ...
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Property of second Steifel-Whitney class of tangent bundle?

Let $M$ be a closed, smooth, simply-connected $4$-manifold. Is $w_2$ the unique class in $H^2(M, \mathbb{Z}_2)$ such that $w_2 \cup x = x \cup x$ for all $x \in H^2(M, \mathbb{Z}_2)$ or not? I ...
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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 ...
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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 ...
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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 $$ ...
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+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.) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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: ...
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1answer
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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 ...
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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, ...
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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) = ...
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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?
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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?
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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.
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1answer
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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 ...
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1answer
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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 ...
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2answers
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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 ...
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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 ...
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$\mathbb{R}^n$ 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?
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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 ...
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1answer
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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 $H_\mu$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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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. ...
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1answer
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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 ...
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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)} $ ...
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$S^n$ 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?
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1answer
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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 ...
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1answer
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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 ...