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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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
1k views

How do different definitions of “degree” coincide?

I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do ...
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2answers
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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 ...
27
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3answers
758 views

Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space

Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- ...
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3answers
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Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
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1answer
2k views

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 ...
25
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10answers
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“Immediate” Applications of Differential Geometry

My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another ...
25
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4answers
723 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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3answers
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Consequences of Degree Theory

I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
21
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1answer
392 views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
20
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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?
20
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2answers
818 views

Question about Milnor's talk at the Abel Prize

I don't quite follow the rough outline Milnor gives of the fact that the 7-sphere has different differentiable structures. The video is available here, and the slides he used can be found here. ...
20
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2answers
539 views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
19
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7answers
970 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 ...
19
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2answers
870 views

Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
18
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2answers
881 views

Relationship between the zeros of a vector field and the fixed points of its flow

I'm having a little trouble here and would appreciate some hints. Let $M$ be a compact manifold without boundary and let $X$ be a smooth vector field on $M$ with only isolated zeros. Let $\theta_t$ ...
18
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3answers
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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 ...
17
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2answers
408 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 ...
17
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1answer
169 views

Reconstructing a manifold from critical points

I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ...
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2answers
365 views

explicit “exotic” charts

can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?
15
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5answers
607 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} ...
15
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4answers
66 views

group operations are smooth in $\text{SL}(n, \mathbb{R})$

I am told the following reason as to why group operations of multiplication and inversion are smooth on $\text{SL}(n, \mathbb{R})$. Multiplication is smooth because the matrix entries of a product ...
15
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2answers
2k 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 ...
15
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2answers
571 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 $\mathbb{R}^3$. ...
15
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2answers
540 views

The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
15
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2answers
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What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
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2answers
531 views

Books on topology and geometry of Grassmannians

Grassmannians are a pretty useful subject in numerous fields of mathematics (and physics). In fact, it was the first non-trivial higher-dimensional example that was given in an introductory projective ...
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1answer
999 views

What does “locally trivial” do for us?

For the following we will work in the smooth category. (But examples in the topological category is also welcome.) The usual definition of a fibre bundle is Def A fibre bundle is the quadruple ...
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567 views

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 ...
14
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3answers
1k 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 ...
14
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1answer
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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 ...
14
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1answer
193 views

Are locally homotopic functions homotopic?

Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such ...
14
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1answer
308 views

Question about the proof of the index theorem appearing in Milnor's Morse Theory

I am trying to get through the proof of the index theorem. The background: I have been stuck for quite a while on the following point which Milnor says is evident: Let $\gamma: [0,1]\rightarrow M$ be ...
14
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1answer
565 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 ...
13
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2answers
880 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 ...
13
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2answers
466 views

Product of spheres embeds in Euclidean space of 1 dimension higher

This problem was given to me by a friend: Prove that $\Pi_{i=1}^m \mathbb{S}^{n_i}$ can be smoothly embedded in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$. The solution is apparently ...
13
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1answer
261 views

Is $\mathbb{S}^\infty$ exotic

During construction of universal bundles one considers (for example) the infinite real projective space $\mathbb{R}\mathbb{P}^\infty$, coming from the sphere $\mathbb{S}^\infty$. My question is, are ...
13
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1answer
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What's so special about a homotopy $15$-sphere?

I just saw a table which counts diffeomorphism classes of homotopy $n$-spheres (that is, spaces homotopy equivalent to $n$-dimensional spheres). Such a table can be seen on the first page of this ...
12
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3answers
244 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
12
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2answers
441 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. ...
12
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2answers
180 views

If a smooth manifold X is covered by an odd sphere, then X is orientable.

In solving some old qualifying exam questions, I've been thoroughly stumped. If a smooth manifold $X$ is covered by an odd dimensional sphere, then $X$ is orientable. I see this question has ...
12
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1answer
412 views

Gradient-like vector fields

Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function. My goal is to better understand gradient-like vector fields for $f$. Question: Do any two ...
12
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1answer
425 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 ...
12
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1answer
399 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 ...
12
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0answers
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+50

Definition of bordism - gluing manifolds with structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U ...
11
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3answers
460 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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5answers
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Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
11
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1answer
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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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2answers
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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 ...
11
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1answer
419 views

uniqueness of the smooth structure on a manifold obtained by gluing

I've just read a proof that If $M$, $N$ are smooth manifolds with boundary and $f: \partial M\rightarrow \partial N$ is a diffeomorphism then $M \cup_f N$ has a smooth manifold structure such that ...