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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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Show that the index of $r$ must be the sum of the indices of $p$ and $q$.

Could someone give me some help to get started with this question? Don't even have the slightest idea.. =( Suppose a vector field v on $\mathbb{R}^n$ has exactly two isolated zeros $p, q$, and $p, q$ are connected by a flow-line of the vector…
WishingFish
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Related to GP 1.3.9 - Is projection function smooth?

I start to think of this question when I attempt Ex 1.3.9 on Guillemin and Pollack's Differential Topology GP 1.3.9(b) Every manifold is locally expressible as a graph.. I am under the impression that this is true. For the proof of the following…
WishingFish
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Relation between Lefschetz number and the degree of a map on the sphere

I would like to check that the Lefschetz number $\Lambda_f$ and the degree $deg_f$ of a smooth mapping $f:\mathbb{S}^2\rightarrow \mathbb{S}^2$ satisfy $$\Lambda_f-deg_f=1$$ but I am clueless. I know that since $\mathbb{S}^2$ is compact, connected…
beginner_user123
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Augument, and injectivity.

I am having much trouble reading the proof Local Immersion Theorem in Guillemin & Pallock's Differential Topology on Page 15. Now we try to augment $g$ so that the Inverse Function Theorem may be applied. How does it mean here to "augement…
WishingFish
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Does $\Gamma$ intersect $SL(2, \mathbb{R})$ transversely at $I$?

Identify the space of all $2 \times 2$ real matrices with $\mathbb{R}^4$ so that the matrix $\left( \begin{array}{cc} a & b\\ c & d\end{array} \right)$ corresponds to $(a, b, c, d)$. Let $\Gamma$ denote the hyperplane in $\mathbb{R}^ 4$ with…
kiwi
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Degree of $f(x)=x^2$

Let $f:I\to I,$ where $I$ is compact set, given by $f(x)=x^2$ First, if $I$ is centered in $0$ then deg$(f)=0,$ cause $f$ isn't onto. But, if we consider $f:I\to f(I),$ then $\operatorname{deg}(f)=2?$ Is true? I thought this way: if $q\in f(I)$ is a…
Kalashinikov
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GP 1.2.2 $T_x(U) = T_x(X) \text{ for } x \in U.$

This is exercise 1.2.2 on Guillemin and Pollack's Differential Topology If $U$ is an open subset of the manifold $X$, check that $$T_x(U) = T_x(X) \text{ for } x \in U.$$ I am fairly confused with this problem, because I found the implicit…
WishingFish
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All 1-tensors are alternating

This statement from page 155 of Guillemin and Pollack's Differential Topology. I would assume because 1-tensors can not alternate because they have nothing to alternate with, so they are alternating...?
1LiterTears
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Show the points $u,v,w$ are not collinear

Consider triples of points $u,v,w \in R^2$, which we may consider as single points $(u,v,w) \in R^6$. Show that for almost every $(u,v,w) \in R^6$, the points $u,v,w$ are not collinear. I think I should use Sard's Theorem, simply because that is the…
1LiterTears
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Tranversal paths between two points

I have been studying differential topology from Hirsch, and sometimes in proofs he takes two points $x,y\in M$ a path between them and then just says that we can assume that this is transversal to a certain submanifold that we are interested in. Now…
Someone
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Jets and their differential structure

I have been reading Hirsch's differential topology and now he introduces the concept of Jets between manifolds. He claims two things If $M$ and $N$ are $C^{s+r}$ manifolds then $J^r(M,N)$ will be a $C^s$ manifold. This I was able to see in an…
Someone
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Difference between the tangent bundle $TM$ with $M\times \mathbb{R}^{\dim M}$

The book I'm using is Lee's Introduction to Smooth Manifolds. I just had my first encounter with the tangent bundle and I'm asked to show that $T\mathbb{S}^1$ is diffeomorphic to $\mathbb{S}^1\times \mathbb{R}$. What is difficult for me is that I…
Shana
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Theorem 1,1, Hirsch Differential topology

Just getting started with Differential topology. Here is the first theorem I struggle with Theorem 1.1. The set $\text{Imm}^r(M,N)$ of $C^r$ immersions is open in $C_S^r(M,N)$, $r \geq 1$. Proof Since $$ \text{Imm}^r(M,N) = \text{Imm}^1(M,N) \cap…
user8469759
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How do smooth manifolds differ from manifolds embedded in $\mathbb{R}^n$?

Instead of defining a smooth manifold to be a manifold whose gluing functions are smooth, what would happen if we defined it as an $n$-manifold $M$ which has an embedding into $\mathbb{R}^{n +1}$? A smooth map between manifolds $e_M : M…
user1172394
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Exercise from Hirsch Differential Topology, Generalization of Fund. Theorem of Algebra

This is problem 11, from Chapter 5 Section 1 (Degrees of Mappings) in Hirsch's differential topology. Let $U \subset \mathbb{R}^n$ be a nonempty open set, and $F :U \to \mathbb{R}^n$ a $C^1$ map. Assume that $F$ is proper (the preimage of a…
rubikscube09
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