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Results tagged with differential-topology
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user 2051
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
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Most general context for the Morse Lemmas
Among the foundational results in differential topology are the Morse lemmas:
Suppose that $f\colon\, M\to \mathbb{R}$ is a smooth function on a closed manifold M, that $f^{−1}[-\epsilon,\epsilon]$ …
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Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$
A fundamental result in three-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of the two-sphere $S^2$ ex …
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Elegant proof that mapping class groups are generated by Dehn twists?
One of the foundational results about mapping class groups of surfaces is that they are generated by Dehn twists. A mapping class is a connected component in a space of diffeomorphisms, so another way …
- 22.2k
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answer
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Is the space of smooth partitions of unity connected? Simply-connected?
One of the requirements for a smooth manifold $M$ is that it be paracompact, and one of the equivalent definitions of paracompactness for a smooth space is that for overy open cover of $M$, there exis …
- 22.2k
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The difference between a handle decomposition and a CW decomposition
Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for $M$, and that it induc …
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How does the Framed Function Theorem simplify Cerf Theory?
A handle decomposition of a manifold $M$ is a useful structure to carry around. It is induced by a Morse function $f\colon\, M\to \mathbb{R}$.
How are two handle decompositions of $M$ related? The sp …
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Searching for an unabridged proof of "The Basic Theorem of Morse Theory"
Steven Smale labels the following statement "The Basic Theorem of Morse Theory" in A Survey of some Recent Developments in Differential Topology:
Let f be a $C^\infty$ function on a closed manifold …
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Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a nuts …
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What do tangles teach us about braids?
A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$ …
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Good introduction to Morse-Novikov theory?
Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued …
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