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Questions tagged [differential-topology]

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.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Let $\deg$ be the topological degree. Then $\deg(fg) = \deg(f)\deg(g)$, with $f, g : M \to N$

Recall that the topological degree is defined as: Let $f : M \to N$ a $C^k$ function and $y$ be a regular value of $f$. Then we define: $$\deg(f)= \sum_{f(x) = y}|Df(x)|,$$ where $| . |$ means the signal, being $1$ or $-1$ depending if $Df(x)$…
L.F. Cavenaghi
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Non-homotopical manifolds with same de Rham cohomology

I am searching for manifolds $M$ and $N$ with different homotopy type such that their de Rham cohomology is isomorphic as rings. It would, of course, be enough to find $M$ and $N$ with different $\pi_1$.
Francisco
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If $f: A \subset \mathbb{R}^m \to \mathbb{R}^n$ is of class $C^1$ and in $a \in A$ rank of $f$ is $p$ there is an embedding

If $f: A \subset \mathbb{R}^m \to \mathbb{R}^n$ is of class $C^1$ and in $a \in A$ rank of $f$ is $p$ there is an embedding $\phi : V \to A$, of class $C^{\infty}$ such that $f\circ \phi$ is an embedding. My attempt was: Let $\mathbb{R}^m =…
L.F. Cavenaghi
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Discrete subgroup of Lie group has properly discontinuous action

I've found some literature which would be helpful if I understood the following, "we can choose neighborhood $U,V$ of the identity such that $VV^{-1} \subset U$ and $U \cap \Gamma = \{e\}$. " Question: I can figure out the last, but how do you…
Fundamental
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why the Poincaré Duality morphism induces a morphism from cohomology to dual cohomology

I am studying the de rham theorem and Poincaré Duality from http://www.few.vu.nl/~vdvorst/DeRham.pdf and I have a question about the Poincaré map \begin{align*} \mathcal{PD} : \Omega^p(M) &\rightarrow\Omega^{n-p}_c(M) ^ { *} \\ \mathcal{PD}…
Werner Germán Busch
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Proving an embedding

Note: below the word embedding is supposed to also be an immersion as in its differential topology definition. Question: $$ f : \mathbb{R}^2 \rightarrow \mathbb{R}, f(x,y) = x^3 + xy + y^3 + 1 $$ for which points among $p1 = (0; 0), p2 = (1/3;…
user2346536
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Equivariant Poincaré Duality

Let $\Gamma$ be a group acting by smooth orientation-preserving diffeomorphisms on a smooth compact oriented manifold $M$ of dimension $n$. The de Rham complex $\Omega^{\bullet}(M)$ of differential forms and the dual complex of currents…
Raphael Ponge
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Openness of homeomorphism in $C^0_S(M,N)$

I'm studying differential topology on Hirsch book, in particular the part on function spaces. There's the proof (page 38) of the fact that the set of $C^r$ diffeomophisms between two $C^r$ manifolds $M, N$ is open in $C^r_S(M,N)$ (here, in page 2,…
popoolmica
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Smooth functions on $\mathbb{R}^k$ as a subset of $\mathbb{R}^l$ are the same as usual.

I am reading a book on differential topology and the first question in it has me confused. If $k < l$ we can consider $\mathbb{R}^k$ to be the subset $\{(a_1, \cdots,a_k, 0, \cdots, 0)\}$ in $\mathbb{R}^l$. Show that smooth functions on…
zrbecker
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Every differentiable structure is smoothable to a smooth structure.

I was studying differentiable manifolds and smooth manifolds. While reading on the Wikipedia website about them, I came across one statement that I have no idea why is it. This statement is Every $C^k$-structure is uniquely smoothable to a…
Hopf eccentric
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Lemma about Brouwer degree in Milnor's book

In Milnor's book "Topology from the Differentiable Viewpoint", He states the following lemma: Let $M, N$ be oriented $n$ manifolds, with $M$ compact and $N$ connected. Let $f : M \to N$ be a smooth map, and $y$ a regular value of $f$. Also, suppose…
Ishan Banerjee
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Smoothness independent of chart

Given a continuous map $f:M_{1}\rightarrow M_{2}$ between differentiable manifolds, a map is smooth if for all $p\in M_{1}$ with there exist charts $\varphi_{1}:U_{1}\rightarrow V_{1}$ and $\varphi_{2}:U_{2}\rightarrow V_{2}$ in $M_{1},M_{2}$…
LCL
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Union of the two coordinate axes in $\mathbb{R^2}$ - From the book '' Differential Topology '' of Guillemin and Pollack

I have tried to solve the problem : Prove that the union of the two coordinate axes in $\mathbb{R^2}$ is not a manifold. Let $X = \{(x,y) \in \mathbb{R^2} : x=0~or~ y=0\}$ be the union of the two coordinate axes in $\mathbb{R^2}$. What happens to a…
user230283
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Topological question from the book '' Differential Topology '' of Guillemin and Pollack

Here's the question : A smooth bijective map of manifolds need not be a diffeomorphism. In fact, show that $$f:\mathbb{R^1}\rightarrow {R^1}$$ $$x\rightarrow f(x)=x^3,$$ is an example. I would like to do this problem, but I'm really not sure I…
user230283
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Commuting smooth maps

Suppose $f:A\to B$ and $g:C\to D$ are smooth embeddings, $h:B\to D$ is a smooth map, and $i:A\to C$ is a continuous map such that $g(i(x))=h(f(x))$. Then, how to show that $i$ is smooth? An embedding f is a smooth map such that the inclusion $X\to…
TROLLHUNTER
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