# Tagged Questions

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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### Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...
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### Poincare lemma for compact vertical supports in Bott & Tu

I'm trying to work out Proposition (6.16) from Bott and Tu which states that $H^*_{cv}(M\times\mathbb{R}^n)$ is isomorphic to $H^{*-n}(M)$. The first group being forms compactly supported along the ...
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### Vector field on n-manifold whose sum of indexes is equal to Euler charasteristic

For 2-manifolds and 3-manifolds such a tangent field (whose singular points indexes sum to manifold's Euler chracteristic) construction can be done visually. For example, for triangulated 2-manifold ...
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### Do smooth compact connected manifolds admit CW compositions with a single 0-cell?

I have stated this question incorrectly before, sorry.
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### On Atiyah-Singer Index theorem and Atiyah-Bott fixed point theorem.

I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic ...
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### Transversality: what is wrong with this counter example to persistence for small perturbations?

Let $M$ and $N$ be differentiable manifolds in $\mathbb{R}^{n}$, and let $p \in \mathbb{R}^{n}$. We say that $M$ and $N$ are transversal at $p$ if $$T_{p} M + T_{p}N = \mathbb{R}^{n}.$$ By dimension ...
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### Do compact connected smooth manifolds admit the structure of a CW complex with a single 1-cell? [closed]

This seems intuitive to me, since they admit a CW decomposition with finitely many cells. But I can't see how to prove it.
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### Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence.

Let $M,N$ be smooth manifolds. It seems to be well known that if the sets $C^0(M,N)$ and $C^\infty(M,N)$ are equipped with the appropriate topologies (I suppose the weak/strong Whitney topology), then ...
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### Degree of smooth map of manifolds depends on orientation choice?

I'm a little to confused as to why it appears that the degree of a smooth map $f: M \to N$ between smooth manifolds appears to only be defined up to sign - I'm not sure where my mistake is. By ...
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### Is there any difference between Immersion and embedding? [closed]

Definition as below , I think they are same ,is right ?
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### Orientation on the boundary

If $M$ is an oriented without boundary manifold, and $\mu$ is it volume form, is true that the boundary of $M\times [0,1]$ is $M \cup M$, right? It is true also that the orientantion on the boundary ...
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### Fundamental class for tangent bundle

For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
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### degree 1 map $f : M \to S^n$

Let us consider $M$ a closed and connected n-manifold which is also orientable. An exercise in Hatcher claims that for any such $M$ there is a continuous map $f: M \to S^n$ such that it's degree is 1, ...
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### Isomorphic modules of sections imply isomorphic bundles

For reference this is about a part of question 3-F in Characteristic Classes by Milnor and Stasheff, which discusses the $C(B)$-module structure of the space of sections of a topological vector bundle ...
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### Every real matrix with non-negative entries has a non negative eigenvalue [duplicate]

If $A$ is any matrix $n\times n$ with non negative entries, then $A$ has a non negative eigenvalue. I know that I have to use the Brower Point fix theorem, but I am not finding the function for that. ...
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### Can every Riemmanian Manifold be completed?

I had two trails of though.. is either of them fruitful? I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's ...
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### How can we define $\partial x_{i_r}^p(X_p^r)$?

Let $M$ be a smooth manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P\subseteq M$ be an open set and $X_1,\ldots,X_s\in\mathcal T(P)$ and $X^1,\ldots,X^r\in\mathcal T^*(P)$ where $\mathcal T(P)$ ...
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### partial differentiation on a manifold

im currently taking up a course on differential geometry and the last topic is topology. id like to ask for help in our homework since im kind of new to this kind of questions which involves proving ...
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### $\operatorname{SU}(n)$ as manifold

I am trying to do this has a while, but I cannot use correctly the regular value theorem to do so! I appreciate any help. The problem is that I cannot choose the function to take $SU_n$ as a regular ...