# 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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### 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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### 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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### Diffeomorphism group of the unit circle

I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and ...
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### 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 ...
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### Is manifold mapping degree equal to algebraic degree for polynomials?

If $M$ and $N$ are oriented $n$-manifolds and $f: M \to N$ then the degree of $f$ is given by $$\deg f = \sum_{p \in f^{-1}(q)} sign_p f$$ where $q$ is a regular value and the sign is $+1$ if $f$ is ...
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### Is a vector bundle orientable if and only if its dual bundle is orientable?

I was reading up on my dual spaces today and I made the following hypothesis: A vector bundle $\xi$ is orientable if and only if $\xi^*$ is orientable. This seems rather intuitive, and although ...
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### Obstructions to lifting a map for the Hopf fibration

This is a bit of an elementary question, but suppose $\pi: \mathbb{S}^3\to \mathbb{S}^2$ is the Hopf fibration, are there reasonably computable obstructions to when a map $f:M\to \mathbb{S}^2$ can be ...
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### How do you imagine the shape of a manifold $S^2 \times S^1$?

In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times. (The following story can be applied not only this manifold but also for any 3-dimensional manifold.) But ...
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### Proof that vector area is a boundary integral?

Let $M \subset \mathbb{R}^2$ be a closed topological disk and let $f: M \rightarrow \mathbb{R}^3$ be a smooth embedding; let $N$ be the corresponding unit normal field on $M$. The vector area is ...
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### 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 ...
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### 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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### 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 ...
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### Morse homology of $P^2$

I have seen and worked through the explicit computation of the Morse homology of the sphere and the torus (with signs and all). But trying it for $\mathbb{R}\mathbb{P}^2$ has lead me to dead ends. Is ...
Let $M$ be a smooth oriented $n$-dimensional manifold and denote by $A \in H_n(M;\mathbb{Z})$ the fundamental class of $M$ (a generator of singular homology consistent with the orientation of $M$). ...
On learning about how to define smooth vector fields on a manifold $M$, I learned that one should first define a tangent bundle , $T(M)$, as $\cup T_p(M)$ together with a topology(smooth structure). ...