Articles of differential topology

Manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$.

Let $H=\{(x, y, z)\in\mathbb{R}^3 | \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^2}=1\}$. Prove that $H$ is a manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$. Already I tried to be spread but not how to begin the diffeomorphism

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like Rolfsen which works with PL knots. I’m really hoping for some grand unifying or almost-unifying theorem. Thank you […]

Immersion, embedding and category theory

The map between smooth manifolds $f:M \to N$ is called an immersion if the tangent map $Tf$ is injective for each $x \in M$. There is a corresponding notion of submersion. About embeddings I met two definitions: the first states that a map $f:M \to N$ is called an embedding if it is an immersion […]

Is a map with invertible differential that maps boundary to boundary a local diffeomorphism?

Let $M,N$ be smooth manifolds with boundary (of the same dimension). Let $f:M \to N$ be a smooth map satisfying $(1) \, \,f(\partial M)=\partial N,f(\operatorname{Int} M)=\operatorname{Int} N$. $(2) \, \, df_p$ is invertible for every $p \in M$. Is it true that $f$ is a local homeomorphism? I suspect $f$ must in fact be a […]

The integral of a closed form along a closed curve is proportional to its winding number

Source: Guillemin-Pollack Exercise 4.8.2. Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 – \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 – \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where $W(\gamma, 0)$ is the winding number of $\gamma$ with respect to the origin. $W(\gamma, 0)$ is defined just like $W_2(\gamma, 0)$, but using […]

Convex combination of projection operators

If $P_1, P_2: V \to V$ are linear projection operators on the vector space $V$ with $R := P_1(V) = P_2(V)$, is it true that any convex combination of $P_1$ and $P_2$ is again a projection operator $P_3$ with $P_3(V) = R$? I’m trying to figure out whether or not the convex combination of two […]

Any two $1$-forms $\alpha$, $\alpha'$ with property satisfy $\alpha = f\alpha'$ for some smooth nowhere zero function $f$?

This is a followup question to here. Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$? Does it follow that any two $1$-forms $\alpha$, $\alpha’$ […]

Is this hierarchy of manifolds correct?

Question: a Hausdorff differentiable manifold (locally Euclidean space): $$ \text{is metrizable} \iff \text{is paracompact} \iff \text{admits a Riemannian metric} \,?$$ Does one also have for a locally Euclidean Hausdorff space (not necessarily differentiable): $$\text{second countable} \iff \text{metrizable with countably many connected components}? $$ Thus second countability is strictly stronger for such spaces than metrizability/paracompactness/existence of […]

Why is the Hopf link the only link with knot group $\mathbb{Z} \oplus \mathbb{Z}$?

We can use the Loop Theorem to show that if $\Sigma$ is a minimal-genus Seifert surface for a link $L$, then $\pi_1(\Sigma)$ injects into the knot group $\pi_1(S^3 \setminus L)$. An orientable connected surface with nonempty boundary and abelian fundamental group must be a disk or an annulus. Therefore $\pi_1(S^3 \setminus L)$ is abelian only […]

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 interval (not necessarily open or closed), and $f$ is smooth (i.e. infinitely differentiable)? The reason I ask is that in my […]