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Results tagged with homotopy-theory
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user 36108
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
1
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Historical transition from classical homotopy to modern homotopy theory
I was re-reading sections of Whitehead's book the other day, and I found it very helpful to think in the way he was writing. For a historical perspective, I would ask Clarence Wilkerson, Peter May, Bi …
7
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Intuition on finite homotopy groups
About $\pi_5(S^4)$, this is already in the stable range, so that it is isomorphic to the cobordism group of immersed curves in the plane. That is clearly ${\mathbb Z}/2$ since you can cancel double po …
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10
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third stable homotopy group of spheres via geometry?
Expanding upon Andre Henriques's answer, $\pi_3^s(P^\infty) \cong {\mathbb Z}/8$. The homotopy group is isomorphic to the cobordism group of non-orientable surfaces in 3-space. Boy's surface is a gen …
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6
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Visualize Fourth Homotopy Group of $S^2$
Let me expand upon jc's comment above. For convenience, I'll pass to the stable homotopy group for a little while. The first stable stem is $\pi_1^s = {\mathbb Z}/(2)$. A representative class is the …
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10
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What do the stable homotopy groups of spheres say about the combinatorics of finite sets?
There are papers by Eccles, Freedman, Koschorke, and Koschorke and Sanderson that date circa 1978-1982 which discuss the $n$th-stable stem as the bordism group of oriented codimension 1 immersions of …
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5
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Embedded ribbons and regular isotopy
A diagram is drawn in the plane. Restrict to knots (not links). Orient the curve, & associate to each crossing a (+/-) via a Right hand rule: palm along over-crossing with pinky pointing towards orien …
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