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Results tagged with homotopy-theory
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user 40804
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.
10
votes
Accepted
Fundamental group of the space of immersions of the 2-sphere in 3-space modulo diffeomorphis...
The action of $\text{Diff}^+(S^2)$ on $\text{Imm}(S^2,\Bbb R^3)$ is free. For pick any immersion $i$ and diffeomorphism $f$; given $x \in \Bbb R^3$ $i^{-1}(x)$ is a closed discrete (because $i$ is an …
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7
votes
Induced maps on homotopy groups by self maps of $\mathbb{CP}^n$
Here is an elementary, if clunky, proof that $|f_*(1)| = d^{n+1}$. I imagine more care with orientations would ensure the sign.
Write $$R:= H^*(\Bbb{CP}^n;\Bbb Z) = \Bbb Z[x]/(x^{n+1}), \;\;\; |x| = …
- 9,580
7
votes
Accepted
Cohomogy of local systems over CW-complexes
I'm not sure of a reference off the top of my head, but the proof is straightforward. Let's say the number of $k$-cells of $M$ is $n_k$.
Because $M$ is given a CW structure, its universal cover $\ti …
- 9,580
147
votes
10
answers
16k
views
What non-categorical applications are there of homotopical algebra?
(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.)
More …
7
votes
Accepted
A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $
I'm not sure exactly what you were claiming, but here is a correct claim.
$\pi_1(N)$ has six order four elements $\{\pm i, \pm j, \pm k\}$, and there are three natural maps $p_1, p_2, p_3: N \to \Bbb{ …
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