# 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.

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### A common misinterpretation of transversality being a stable property.

In the first Chapter in Differential Topology by Guillemin Pollack (Section 6:Homotopy and Stability),the authors have asserted that, Transversality is a stable property in the sense that if $X$ is a ...
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### In how far is the Lyapunov spectrum characteristic for a dynamic system?

If two dynamic systems have the same Lyapunov spectrum on their respective attractor (of equal dimension, of course), which results relate to the properties of these two dynamic systems on those ...
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Let $\lambda\in\mathbb{R}^{+}\setminus\mathbb{Q}$ and $C\in(\mathbb{R}\setminus\{0\})$. Consider \Sigma=\{(x_1 , x_2 , y_1 , y_2)\in\mathbb{R}^{4} : y^{2}_{1}+y^{2}_{2}=C^{2}(x^{2}_{1}+x^{2}_{2})^{\...
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### Relation between two theories of degree of maps

For a holomorphic map between 2 Riemann surfaces, we can define its degree to be the sum of ramification index of all preimages of a point, or simply the cardinality of the preimage of a regular(not ...
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On page 78 of the book Differential Topology by Guillemin and Pollack is presented the following Theorem: If $f_0 , f_1 : X\to Y$ are homotopic and both transversal to $Z$, then $I_{2}(f_0 , Z)=I_{2}(... • 645 0 votes 0 answers 17 views ### Transversality for Morse homology on manifold with boundary I'm trying to make my own argument for a situation where$X$is a smooth manifold with boundary and$f$a Morse function. The vector field$V=-grad f$and we denote$D_p, A_q$as the unstable manifold ... 0 votes 0 answers 44 views ### Pre image orientation and smooth map I consider a smooth map$ F: X\to N$where$X$and$N$are smooth, oriented manifolds of dimension$m+1$and$m$respectively,$X$being compact and with boundary$\partial X=M$. For a regular value$...
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Let $M$ be an n-dimensional smooth manifold (Open or compact). I want to know if it is possible to construct an smooth vector field with exactly one singularity, such that the set of periodic integral ...