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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If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.

Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using ...
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Which manifolds are parallelizable?

Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
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concrete examples of tangent bundles of smooth manifolds for standard spaces

I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$ Do the tangent bundles of the following ...
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Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?

Let $M=\{(x,|x|): x \in (-1, 1)\}$. Then there is an atlas with only one coordinate chart $(M, (x, |x|) \mapsto x)$ for $M$. We don't need any coordinate transformation maps to worry about ...
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262 views

fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
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1answer
283 views

The Hairy Ball theorem and (non-orientable) real projective plane

Is it possible to prove the Hairy Ball theorem via non-orientability of $P^2(\mathbb{R})$? That is, the non-vanishing section $s \colon S^2 \to TS^2$ would induce (via “2-to-1” bundle $p \colon S^2 ...
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410 views

Immersive Injections whose images are Embedded Submanifolds

Let $M,N$ be smooth manifolds where the dimension of $M$ is less than or equal to the dimension of $N$. Suppose that $F: M \rightarrow N$ is an injective immersion and $F(M)$ is an embedded ...
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975 views

Why maximal atlas

This has been on the back of my mind for one whole semester now. Its possible that in my stupidity I am missing out on something simple. But, here goes: Let $M$ be a topological manifold. Now, even ...
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precise official definition of a cell complex and CW-complex

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and ...
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444 views

No hypersurface with odd Euler characteristic

Here is a classic problem which I encountered and could not solve: Prove that a simply connected closed smooth manifold has no closed sub-manifold of co-dimension $1$ with odd Euler ...
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500 views

$f:\mathbb{S}^1\rightarrow\mathbb{S}^1$ odd $\Rightarrow$ $\mathrm{deg}(f)$ odd (Borsuk-Ulam theorem)

I'm having trouble understanding the proof of Borsuk-Ulam theorem ($n=2$) that we did in our class. The only problematic part is the last sentence in the proof of lemma 1. ...
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455 views

embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$

Consider the classic map $$F:\mathbb{RP}^2\rightarrow \mathbb{R}^4$$ defined by $$F[x,y,z]=(x^2-y^2,xy,xz,yz)$$. This defines a smooth embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$. It is clearly ...
7
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207 views

normal bundle of level set

Let $M$ be a Riemannian manifold and $S \subset M$ a regular level set of a smooth function $f:M\rightarrow \mathbb{R}^k$. How can I show that the normal bundle of $S$ is trivial? If $k=1$ then ...
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234 views

Understanding Hom-bundle on vector bundles (Atiyah)

I am trying to make sense of the following statment (from Atiyah's K-theory book) Suppose $V,W$ are vector spaces, and that $E=X \times V, F=X \times W$ are the corresponding product bundles. Then ...
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608 views

Compact submanifolds of $\mathbb{R}^n$ without boundary

I'm having a little trouble seeing how to do Exercise 7.5 in Lee Smooth Manifolds: Let $M$ be a smooth compact manifold. Show there is no submersion $F:M\rightarrow\mathbb{R}^k$ for any $k>0$. ...
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947 views

How to apply Stokes' Theorem for manifolds with boundary

Original motivation: How can I apply Stokes' Theorem to the annulus $1 < r < 2$ in $\mathbb{R}^2$? Concerns: Since the annulus is a manifold without boundary, it would seem that Stokes' ...
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311 views

Can any smooth manifold be realized as the zero set of some polynomials?

Is any real smooth manifold diffeomorphic to a real affine algebraic variety? (I.e. is there an "algebraic" Whitney embedding theorem?) And are all possible ways of realizing a manifold $M$ as an ...
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Showing that a level set is not a submanifold

Is there a criterion to show that a level set of some map is not an (embedded) submanifold? In particular, an exercise in Lee's smooth manifolds book asks to show that the sets defined by $x^3 - y^2 = ...
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1answer
427 views

Elucidating Tu's Definition of a Regular Submanifold

Definition 9.1. A subset $S$ of a manifold $N$ of dimension $n$ is a regular submanifold of dimension $k$ if for every $p\in S$ there is a coordinate neighborhood $(U,\phi) = (U,x^{1},...,x^{n})$ ...
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258 views

Exercise concerning the Lefschetz fixed point number

I can't see a good approach to the third part of the following problem: Let $f: M \to M$ be a smooth map of a compact oriented manifold into itself. Denote by $H^q(f)$ the induced map on the ...
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2answers
650 views

Examples of compact, nonorientable n-manifolds

Among the best-known examples of nonorientable, compact manifolds are projective spaces. However for these one has the fact that $\mathbb RP^n$ is orientable iff $n$ is odd, so that only "half" of ...
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1answer
182 views

Showing a hypersurface is contained in a level set of a regular value

I'm stuck on the following problem: let $S$ be a compact orientable hypersurface in the symplectic manifold $(M,\omega)$. Prove that there exists a smooth function $H: M \to \mathbb R$ such that $0$ ...
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484 views

Examples of Computations in Algebraic Topology

I have started reading "Differential Forms in Algebraic Topology" by Bott, Tu, recently. While I'm quite happy with the exposition of the theorems and explanation of theoretical results, I'm missing ...
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159 views

Invariance of Wall's self-intersection under the regular homotopy

For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in ...
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Question on the transversality between sections

Let $M^n$ be a differentiable manifold and $\pi\colon E\to M$ is $n$-dimensional vector bundle over $M$. We have a zero section $s\colon M\to E$ of $\pi$. How can I make a section $s'$ which is ...
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0answers
115 views

Question on the transversality

Let $f\colon N^n\to M^{2n}$ be an immersion. Then, we can extend $f$ to $\bar{f}\colon E(\nu_g)\to M$ of the total space of the normal bundle. Let $s_0\colon N\to E(\nu_g)$ be a zero section and ...
4
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1answer
279 views

Lifting of a tangent bundle

I have a problem with Kuranishi's theorem in deformation theory. I'll try to formulate it in general terms, and then describe the particular situation. Let $\pi : M \to S$ be a smooth fiber bundle - ...
5
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1answer
648 views

Classification of lens space

Let $L(p,q)$ be the lens space, that is $L(p,q)=S^3/\mathbb{Z}_p$. Here, $\mathbb{Z}_p$ acts on $S^3$ by $(z_1,z_2)\mapsto (\rho z_1,\rho^q z_2)$, $ \rho=e^{\frac{2\pi i}{p}}$. It is well known ...
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588 views

The Strong Whitney Embedding Theorem-Any Recommended Sources?

Just about all of the standard textbooks on manifold theory give proofs of weak versions of the Whitney Embedding theorem. But other then Whitney's original 1944 paper,are there any standard sources ...
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180 views

On the immersion, regular value theorem

Let $g\colon N\to M$ be an immersion. Then, I think that $g^{-1}(p)$ is finite set or $0$-dimensional manifolds for all $p\in M$. Now, let $g_t\colon N\to M$ be an one-parameter family of an ...
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2answers
227 views

Degree of a map between product of manifolds

Let $M^m$ and $N^n$ be compact, oriented smooth manifolds without boundary. Then what is the degree of the map $$ f: M\times N \to N \times M$$ given by $f(x,y) = (y,x)$? I have the feeling it ...
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1answer
304 views

The mod 2 degree of a function when the image space N has a boundary

I was flipping through Milnor's "Topology from the Differentiable Viewpoint," and I came upon a sentence concerning the mod 2 degree of a function from M to N. It essentially says: "We may as well ...
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966 views

Cohomology of complex projective plane

How can I compute Cohomology of complex projective plane $CP^2$? Any magic like the one here?
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574 views

Applications of Morse theory

Background The use of tools from algebraic topology to study simplicial complexes coming from point cloud data has been thoroughly discussed in the papers of Carlsson, Zomorodian, Ghrist, ...
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1answer
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Cohomology of projective plane

How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
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explicit “exotic” charts

can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?
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453 views

Different definitions for submanifolds

I'm trying to better understand the concept of differentiable submanifold. However, it looks like many different definitions are adopted by various authors and so I'm trying to keep myself in sync by ...
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7answers
1k views

Why abstract manifolds?

If we can use Whitney embedding to smoothly embed every manifold into Euclidean space, then why do we bother studying abstract manifolds, instead of their embeddings in $\mathbb{R}^n$? A vague ...
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Visualising a specific orbifold

Let $1 < k \in \mathbb N$ and $M = \{(z_1, z_2) \in \mathbb C^2 : k|z_1|^2 + |z_2|^2 = 1\}$. Let $S^1$ act on $M$ via $e^{i\theta}(z_1,z_2) = (e^{ik\theta} z_1, e^{i\theta} z_2)$. Then I am told ...
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Smooth Poincaré Conjecture

One of my professors wrote the following open question on the blackboard: If $M$ is a compact, connected smooth $4$-manifold such that $\pi_1(M) = 0$, $\pi_2(M) = 0$ (first two homotopy groups are ...
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3answers
2k views

Is every embedded submanifold globally a level set?

It's a well-known theorem (Corollary 8.10 in Lee Smooth) that given a smooth map of manifolds $\phi:M\rightarrow N$ and a regular value $p\in N$ of $\phi$, the level set $\phi^{-1}(p)\subset M$ is a ...
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3answers
467 views

What is $T\mathbb{S}^2$?

I recently learned that the only parallelizable spheres are $\mathbb{S}^1$, $\mathbb{S}^3$, and $\mathbb{S}^7$. This led me to wonder: What is $T\mathbb{S}^2$? Is it diffeomorphic to a more ...
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Trying to draw the tautological line bundle ($\subseteq \mathbb{CP}^1\times \mathbb{C}^2$)

In order to learn about vector bundles, I would like to draw the tautological vector bundle over the complex projective line $$ E = \{(x,v) \in \mathbb{CP}^1 \times \mathbb{C}^2 : v \in x \} .$$ ...
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How to draw a complex line bundle?

The most basic example of a topologically non-trivial real line bundle is the well-known Möbius strip. Everyone who learns about vector bundles will be confronted by it, if only because it has the ...
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Manifold with 3 nondegenerate critical points

Suppose $M$ is a n-dimensional (compact) manifold and $f$ is a differentiable function with exactly three (non-degenerate) critical points. Then one can show, using Morse theory, that $M$ is ...
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640 views

Orientation induced on submanifolds

Suppose you are given two oriented manifolds with boundary $M$ say $B, B'$ and $\partial B = M = \partial B'$. Identify the boundaries and form $C = B \sqcup_{Id: M \to M} B'$. I want to ...
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1answer
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Compact manifold/Morse theory

I have a question concerning the proof of theorem 3.5 in Milnor's Morse Theory. This theorem states that if $f$ is a differentiable function on a Manifold M with no critical points, and if each $M^a ...
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1answer
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Inverse of regular value is a submanifold, Milnor's proof

In Milnor's famous book "Topology from the Differential Viewpoint" he proves the following on page 11: If $f: M\to N$ is a smooth map between manifolds of dimension $m\geq n$ and if $y\in N$ is a ...
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Is every Compact $n$-Manifold a Compactification of $\mathbb{R}^n$?

I read the result that every compact $n$-manifold is a compactification of $\mathbb{R}^n$. Now, for surfaces, this seems clear: we take an n-gon, whose interior (i.e., everything in the n-gon except ...
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1answer
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Diffeomorphic, group-isomorphic Lie groups that are not isomorphic as Lie groups

Do there exist two Lie groups which are diffeomorphic as smooth manifolds, have isomorphic group structures, yet are not isomorphic as Lie groups? Of course, for this to happen, any diffeomorphism ...