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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Properties of Surgery on Manifolds?

I am trying to give a brief explanation in which I make use of the concept of surgery on an $m$-manifold $M$. This is along the lines of (and taken generously from) the Wikipedia entry on Surgery; ...
2
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0answers
159 views

Topology of pseudo projective space

I don't know if "pseudo projective space" is a general accepted term, but I once read a book on general topology where the term was used for $\mathbb{S}^n / (\mathbb{Z}/m\mathbb{Z})$ (where you get ...
3
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1answer
208 views

4-Manifolds of which there exist no Kirby diagrams

In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin ...
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4answers
157 views

holomorphic exoticness

A topological manifold is an exotic copy of another smooth manifold if it is homeomorphic to it, but not diffeomorphic (and when you switch diffeomorphic by homotopic, you get a fake copy, following ...
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144 views

non-orientable 4-manifolds

Most of the books and texts I read about classfication problems surrounding 4-manifolds which are closed and orientable (with a occasional side-track to open orientable 4-manifolds). This is ...
3
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2answers
125 views

Looking books about the topology of n-manifold ($n > 4$)

There are a lot of books dealing with the strangeness of the topology of 4-dimensional topology. I wonder if there are books or overview references on the topology of n-manifolds (where n > 4) ? ...
4
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1answer
584 views

Bundle orientability vs manifold orientability

Given a vector bundle, I am a bit hazy about the difference between the notions of its orientability as a bundle and as a manifold. I think I know that the following are true, A tangent bundle of a ...
6
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1answer
154 views

Exotic Manifolds from the inside

As we know, an exotic $\mathbb{R}^4$ is a manifold which is homeomorphic, but not diffeomorphic to the standard $(\mathbb{R}^4,id)$, and there are even very explicit descriptions of them (Kirby ...
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2answers
332 views

is any subset of a manifold a submanifold?

by definition a submanifold is a subset of a manifold which is itself a manifold. consider $A$ a subset of an $n$-manifold $M$. a neighborhood of $x\in A$ is $\mathbb R^n$ since $x$ is an element of ...
2
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1answer
134 views

Is 2x (360°) rotated (orientable) Möbius band homeomorphic to $S^1\times[0,1]$?

I'm thinking yes, because they are both a quotient of the square. But I can't figure out what the actual homeomorphism is. Do we have to "go outside of $\mathbb{R}^3$" with the homeomorphism? ...
10
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2answers
478 views

Gradient nonzero extensions of a vector field on the circle

Let $\mathbf{v}=(a,b)$ be a smooth vector field on the unit circle $\mathbb{S}^{1}$ such that $a^{2}+b^{2}\neq0$ everywhere in $\mathbb{S}^{1}$ with degree $\deg\mathbf{v}=0$. Suppose also that ...
3
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1answer
260 views

Sphere eversion and Smale-Hirsch theorem

For two manifolds $M^m$ and $N^n$ with $m<n$ the Smale-Hirsch theorem says that the differential map $d:\operatorname{Imm}(M,N)\to\operatorname{Mon}(TM,TN)$ is a weak homotopy equivalence, where ...
5
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1answer
250 views

Fiber of jet bundle of a fiber bundle

Given a fiber bundle $p:E\to B$ with fiber $V$ and structure group $G$, one can define the corresponding $k$-jet bundle $E^k\subset J^k(B,E)$ of jets of local sections of $E$. On Wikipedia there is a ...
2
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2answers
96 views

a neighborhood of an intersection point

if a point $x$ is in the intersection of two spaces $X$ and $Y$ suppose we know explicitly a neighborhood of $x$ in $X$, can we take the same neighborhood of $x$ in $Y$. More specifically, if the ...
18
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2answers
808 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 $\theta_t$ ...
3
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1answer
116 views

The prime decomposition of 3-manifold

Let $Y^3$ be a closed 3-manifold with $\pi_1(Y)=\mathbb{Z}$. Is it true that $Y$ is homeomorphic to $S^1\times S^2$ from the prime decomposition of 3-manifold?
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3answers
1k views

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 ...
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1answer
206 views

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 ...
29
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2answers
2k views

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 ...
16
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3answers
942 views

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 ...
14
votes
2answers
535 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 ...
2
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0answers
240 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 ...
0
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1answer
256 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 ...
4
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1answer
351 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 ...
13
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2answers
844 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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1answer
2k views

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 ...
17
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2answers
395 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 ...
11
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2answers
401 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. ...
8
votes
1answer
357 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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2answers
171 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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1answer
195 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 ...
7
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1answer
447 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 $k>0$. ...
9
votes
1answer
823 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' ...
5
votes
1answer
267 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 ...
8
votes
2answers
918 views

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 = ...
2
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1answer
356 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 $(U,\phi) = (U,x^{1},...,x^{n})$ ...
4
votes
1answer
235 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 ...
2
votes
2answers
555 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 ...
2
votes
1answer
152 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 $0$ ...
8
votes
2answers
441 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 ...
5
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0answers
148 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 ...
2
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0answers
109 views

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 ...
2
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0answers
107 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
votes
1answer
267 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
votes
1answer
474 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 ...
4
votes
2answers
518 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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votes
1answer
172 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 ...
2
votes
2answers
216 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 ...
4
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1answer
258 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 ...
5
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2answers
810 views

Cohomology of complex projective plane

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