Mathematics Stack Exchange
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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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What is the difference between Tangent Bundles and Trivial Vector Bundles.

Tangent bundle: $TM := \bigcup_{p \in M} T_pM$, where $T_pM = \{p\} \times \mathbb{R}^n$. So, $M$ is an $n$ dimensional manifold. Now, letting $V = \mathbb{R}^n$. A trivial vector bundle is $E := M \times V$. I know that this is trivial because we…
Jean Valjean
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What is wrong with this proof (that all vector bundles of the same rank are isomorphic)?

Suppose I have two vector bundles $E \rightarrow M, E' \rightarrow M$ of rank $k$ on a smooth manifold $M$. Let $\mathcal{E}(M), \mathcal{E'}(M)$ denote their spaces of smooth sections. We can choose some arbitrary isomorphism $\phi_p: E_p…
Student1235
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Locally Euclidean can be defined for whole of $\mathbb{R}^n$, or an open set or open ball of $\mathbb{R}^n$?

Topological manifolds are defined to be locally Euclidean (e.g. John Lee). That is, any point is in an open set that is homeomorphic to either $\mathbb{R}^n$, an open ball in $\mathbb{R}^n$ or an open subset of $\mathbb{R}^n$. I understand why…
Jean Valjean
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Confusion about Poincaré-Hopf

The two following theorems appear to be contradictory. Both are proved in Milnor's "Topology from the differentiable viewpoint." I'm sure I'm overlooking something incredibly trivial: Let $f$ be a smooth map from a smooth compact oriented manifold…
Tim kinsella
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If $F$ is smooth then the pushforward $F_*$ is smooth

Question: Suppose $F:M\to N$ is a smooth map. Show that $F_*:TM\to TN$ is a smooth map. We need to find charts $(U,\phi)\subset TM$ and $(V,\psi)\subset TN$ such that $\phi\circ F_*\circ\psi^{-1}$ is smooth. Since for any vector $r$, $\psi^{-1}(r)$…
figura
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De Rham Cohomology Question

Let $M$ be a compact connected $n$-manifold without boundary. Let $\mu\in\Omega^{n-1}(M)$, show that there exists a point $p\in M$ such that $d\mu(p)=0$.
zeroprobability1
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computing intersection number via differential forms

Let $M$ be a $2n$ dimensional manifold, and let $A$ be a $n$-dimensional cycle on $M.$ I want to compute the self-intersection $(A.A)$ of $A$ with itself. Let $\eta_A$ be the form in $H^n(M, \mathbb{R})$ given by the Poincare duality isomorphism,…
ymo
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Compute $d\omega$ and $\int_{S^2}\omega$.

I am wondering if my solution is correct? Thanks. (a) On $\mathbb{R}^3$, let $\omega = y dx \wedge dz.$ Compute $d\omega$ and $\int_{S^2}\omega$, where $S^2$ is the unit sphere centered at the origin, oriented as the boundary of the ball. Revised…
1LiterTears
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dimension of total space of a smooth vector bundle

I'm reading John Lee's Introduction to Smooth Manifold Chapter 10. I'm thinking about the definition of the vector bundle arising from the use of dimension in a particular problem. Suppose we are given a smooth vector bundle $(E,M,\pi)$ of rank $k$,…
Anthony
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Let $M$ be a smooth simply-connected compact manifold of dimension $n$. Is there an immersion of $M$ into the $n$-torus $T^n = S^1 × . . . × S^1?$

I do not know how to do the following qualifying exam problem. Any helped is nice. Let $M$ be a smooth simply-connected compact manifold of dimension $n$. Is there an immersion of $M$ into the $n$-torus $T^n = S^1 × . . . × S^1?$
okipik
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Embedding of the cotangent of the n sphere in R^2n

It is true that $T^* S^1$ embeds in $\mathbb{R}^2$ as the cotangent of $S^1$ is trivial and $S^1 \times \mathbb{R}$ is diffeomorphic to the punctured plane. Is it true in general that $T^* S^n$ embeds in $\mathbb{R}^{2n}$ ?
Vincent L.
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Linearity of push forward $F_*$

How can I prove the linearity of $F_*$? What does $F_*$ eat? If $N$ is smooth manifolds and $F: M \to N$ is a smooth map, for each $p \in M$ we define a map $F_*: T_pM \to T_{F(p)}N$, called the push-forward associated with $F$, by $$(F_*X)(f) =…
Tumbleweed
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The degree of antipodal map.

I am trying to solve the problem A map without fixed points - two wrong approaches. But I am not certain about the degree of antipodal map. I my thought, since the preimage of a point $y \in S^k$ is just $-y$, the degree is just $+1$ or $−1$,…
WishingFish
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A map without fixed points - two wrong approaches

For the unit sphere $S^n \subset \mathbb{R}^{n+1}$ let $f : S^n \to S^n$ be the map reversing the signs of all but one coordinate, $$f(x_0, x_1, \dots, x_n) = (x_0, -x_1, \dots, -x_n):$$ (a) Compute the Lefschetz number $L(f)$. My attempt to this…
WishingFish
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Derivative of a differential

Let $\rho_t$ be the one-parameter flow associated to a vector field $v$ on a manifold $M$. What is $\frac{d}{dt} d_p\rho_t$? Intuitively, $$ \frac{d}{dt} d_p\rho_t = d_p \frac{d}{dt} \rho_t = d_p (v \circ \rho_t) = d_{\rho_t(p)}v \cdot d_p…
mathandchess
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