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Questions tagged [differential-topology]

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.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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morse function and immersion

I have a question from Pollack and Guillemin's book: let $X$ be a smooth manifold. $f:X\to \mathbb R^n$ an immersion, $f=(f_1,....f_n)$. show that for almost all $a_1,...,a_n$, the function $a_1f_1+.....a_nf_n$ is a morse function on $X$. I know so…
user56714
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Evaluate the index of a vector field at an isolated zero, using the Brouwer degree

I want to find the index of v(x,y)=(x,-y) at (0,0), which is equal to the degree of the mapping $f:S^1\to S^1$ defined by $$f(x,y)=\frac{v(x,y)}{||v(x,y)||}.$$ To find $\deg(f)$, one needs to pick a regular value $\mathbf y$ and…
Boar
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Local diffeomorphism of Euclidean space is a global diffeomorphism

I am learning about the Legendre transform and found in Mac Lane's Geometrical Mechanics lecture notes (v1 p.54) the following inversion theorem: The Legendre transformation $\ v\in V\mapsto dL(v)\in V^*$ is invertible if and only if the Hessian…
jpdm
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Implicit function theorem to prove dimension invariance of diffeomorphisms?

Basic question. I started reading Ordinary Diff. Equations by V. I. Arnold and am a little confused about one of the exercises: proving a diffeomorphism from U to V (in the context of the text, this is defined as a one-to-one invertibly…
devr
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Example in the proof of Sard Theorem

In the book Topology from the differential viewpoint - Milnor, there is a proof of Sard Theorem : When $f:X^{n+m}\rightarrow Y^n$ is a smooth map where $X^N$ has $N$ dimension, then $f(C^1)$ has a measure $0$ where $C^1 =\{ x\in X |df_x=0\}$. Here…
HK Lee
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Compact Manifold with odd dimension has Euler numer 0. (Proof using differential topology)

So I am trying to prove this, as an excercise for a course in differential topology, the previous part of the excercise was to find $\chi(S^n)$, which is $1+(-1)^n$, it is likely that this fact is meant to use in this part. I really do not know how…
Bajo Fondo
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What am I doing wrong? Exercise 2, chapter 2, section 3 from Guillemin and Pollack.

I am doing exercise 2, chapter 2, section 3 from Guillemin and Pollack's ''Differential Topology''. Part of the excercise is to prove that given a compact manifold $Y \subset \mathbb{R}^m$, and a pont $w \in \mathbb{R}^m$, there exist a point (not…
Bajo Fondo
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A vector field corresponding to the complement of the tangent bundle

Let $M$ be a $m$ dimensional orientable manifold, and $N$ a $m-1$ dimensional orientable submanifold in $M$, then we know at each point $x \in N$, $T_{x}M = T_x N \oplus$its complement. I need to produce a vector field $X$ such that $X_x$ is a…
Keith
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Is $S^2 \times S^2$ diffeomorphic to $S^1 \times S^3$?

I am trying to analyze whether $S^2 \times S^2$ are diffeomorphic to $S^1 \times S^3$. First of all, the dimension matches because they are all four-dimensional manifolds. Then I tried thinking about techniques to prove whether manifolds are…
penny
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Deciding whether a set is a smooth submanifold

I ma given the following sets $X_a = \{(x,y,z) \in \mathbb{R}^3 | ax = y^2 + z^2\}$ and $X_b = \{(x,y,z) \in \mathbb{R}^3 | y^2 + bz= 1\}$. I am asked for which values of $a$ and $b$ are $X_a$ a smooth 2-manifold in $\mathbb{R}^{3}$, for which…
BOlivianoperuano84
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Local submersion theorem G&P

Guillemin and Pollack section 1.4, page 21, corresponding figure 1-12. Select local coordinates around $x$ and $y$ so that $f(x_1, ...x_k) = (x_1, ...., x_l)$ and $y$ corresponds to $(0, ..,0)$. Thus near $x$, $f^{-1}(y)$ is just the set of points…
G. Debailly
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Stability and Morse functions.

I'm studying differential topology, I know that there are many properties which are stable, but I don't know if to be a Morse function is a stable property. I think that it is not true, but I haven't found a counterexample. Thanks!
Alexandre Muñoz
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2.1.11 Guillemin and Pollack

Let $X$ be a manifold with boundary. For each point $x\in \partial X$, we have a neighborhood in $X$, say $U_x$, which is diffeomorphic $\phi_x:U_x\to H^k$ to $k-$dimensional upper half space. Then let $\pi:U_x\to\mathbb{R}$ be projection onto the…
J. Moeller
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If $k < l$ show that smooth function on $\mathbb{R}^k$ considered as a subset of $\mathbb{R}^l$ are the same as usual.

If $k < l$ we can consider $\mathbb{R}^k$ to be the subset $\{(a_1, ..., a_k, 0, ..., 0)\}$ in $\mathbb{R}^l$. Show that smooth functions on $\mathbb{R}^k$ considered as a subset of $\mathbb{R}^l$ are the same as usual. There are two definitions…
Perturbative
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Proving finiteness of a submanifold

For some reason I am having trouble parsing this bit from Guillemin & Pollack Chapter 2.4: Let $X, Z$ be transversal closed submanifolds in $Y$ (everything is without boundary). Further, let $X$ be compact. Then, $X \cap Z$ is compact and…
Apoha
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