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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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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1answer
171 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
210 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
256 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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2answers
798 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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459 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
1k views

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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350 views

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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1answer
386 views

Different definitions for submanifolds

this is my first time here! 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 ...
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7answers
904 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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1answer
125 views

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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1answer
287 views

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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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 point $p\in N$ of $\phi$, the level set $\phi^{-1}(p)\subset M$ is a ...
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447 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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3answers
500 views

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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2answers
628 views

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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2answers
278 views

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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2answers
567 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
269 views

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
747 views

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 ...
24
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4answers
694 views

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
362 views

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 ...
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1answer
136 views

Simple Continuity Question concerning Vector Bundles

I'm a bit confused about the following part in Sir Michael Atiyah's "K-Theory." Let $E = X \times V$ and $F = X\times W$, and let $\phi: E\to F$ be a vector bundle homomorphism. Why is the induced ...
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5answers
581 views

How should I think about what it means for a manifold to be orientable?

Let M be a smooth manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that for all $\alpha, \beta$, $\textrm{det}(J(\phi_{\alpha} ...
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446 views

Intuition for not-so-smooth manifolds

in standard text books on (smooth) manifolds, for example the known series by John M. Lee or Jeffrey Lee, you either deal with continuous manifolds, or with smooth manifolds. However, neither in ...
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1answer
123 views

Subbundles and subsheaves

Let Let $E \rightarrow X$ be a vector bundle on a manifold $X$. Let $\cal E$ be the sheaf of sections of $E$. Let $\cal F$ be a subsheaf of $\cal E$, and let $F$ be the etale space of $\cal F$. What ...
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8answers
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Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ...
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2answers
536 views

Why study “curves” instead of 1-manifolds?

In most undergraduate differential geometry courses -- I am thinking of do Carmo's "Differential Geometry of Curves and Surfaces" -- the topic of study is curves and surfaces in $\mathbb{R}^3$. ...
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1answer
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Is there an easy way to show which spheres can be Lie groups?

I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it ...
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2answers
306 views

Examples of manifolds that cannot be embedded in $\mathbb R^4$

Could someone give me an example of a (smooth) $n$-manifold $(n=2, 3)$ which cannot be embedded (or immersed) in $\mathbb R^4$? Thanks in advance! S. L.
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1answer
358 views

Diffeomorphisms and Stokes' theorem

Problem: Let $\omega\in\Omega^r(M^n)$ suppose that $\int_\sum \omega = 0$ for every oriented smooth manifold $\sum \subseteq M^n$ that is diffeomorphic to $S^r$. Show that $d\omega = 0$. Proof: ...
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2answers
256 views

Do all manifolds have a densely defined chart?

Let $M$ be a smooth connected manifold. Is it always possible to find a connected dense open subset $U$ of $M$ which is diffeomorphic to an open subset of R$^n$? If we don't require $U$ to be ...
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1answer
519 views

Compactification of Manifolds

It is known that for any locally compact Hausdorff space X, we can define a Hausdorff one-point compactification containing X. In the case of the (differentiable) manifold $\mathbb R^n$ this one-point ...
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3answers
583 views

Does having a codimension-1 embedding of a closed manifold $M^n \subset \mathbb{R}^{n+1}$ require $M$ to be orientable?

I'm trying to follow a proof about immersing/embedding $\mathbb{RP}^n$ into $\mathbb{R}^{n+1}$, which goes roughly as follows: Write $\tau=T\mathbb{RP}^n$. The normal bundle $\nu$ has rank 1, so its ...
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1answer
198 views

Effect of curvature of spacetime on intrinsic geometric properties (under general relativity)

If a space has curvature, then the curvature can be seen intrinsically by finding sums of angles in triangles made of geodesics. Under general relativity, space-time is curved on local scales. On ...
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3answers
431 views

Proper maps and families of compact complex manifolds

Kodaira defines a complex analytic family of compact complex manifolds as the data $(E,B,\pi)$, where $E$ and $B$ are complex manifolds, and $\pi$ is a surjective holomorphic submersion such that the ...
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1answer
283 views

Is the notion of density really needed to define integration on nonorientable manifolds?

I am trying to understand, in as simple terms as possible: How to define integration for non-orientable manifolds, and why it is impossible to do so using only differential forms. In particular, ...
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2answers
638 views

Classification Theorem for Non-Compact 2-Manifolds? 2-Manifolds With Boundary?

I recently learned about the Classification Theorem for compact 2-manifolds. Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones? Moreover, is there a theorem ...
7
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1answer
281 views

When does the topological boundary of an embedded manifold equal its manifold boundary?

Suppose I embed a manifold-with-boundary $M$ in some $\mathbb{R}^n$. Are there conditions (necessary, sufficient, or both) that can help determine when the topological boundary of $M$ is equal to the ...
3
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1answer
197 views

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 $H^n(M, \mathbb{R})$ ...
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1answer
187 views

Which smooth 1-manifolds can be represented by a single smooth parametrization?

Among the smooth 1-manifolds (with or without boundary) which embed into $\mathbb{R}^2$, which ones can be represented by a single parametrization $z = (x,y) = f(t)$, for $t \in I$, where $I$ is an ...
4
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2answers
492 views

Parallel transport of a vector along two distinct curves

Let $\mathcal{M}$ be an n-dimensional manifold endowed with an affine connection $\nabla$. Let $\gamma_1:[a,b]\rightarrow M$ and $\gamma_2:[c,d]\rightarrow \mathcal{M}$ be two curves with the same ...
5
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1answer
730 views

Converse To Quotient Manifold Theorem [Exercise in Lee Smooth Manifolds]

I would like help with the following problem (chapter 9, #4) from Lee's Smooth Manifolds [its not homework, I'm reading it and I got stuck on this one] If a Lie group $G$ acts smoothly and freely on ...
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2answers
1k views

Precise connection between Poincare Duality and Serre Duality

The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from ...
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288 views

Which continuous functions are polynomials?

Suppose $f \in C(\mathbb{R}^n)$, the space of continuous $\mathbb{R}$-valued functions on $\mathbb{R}^n$. Are there conditions on $f$ that guarantee it is the pullback of a polynomial under some ...