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Results tagged with homotopy-theory
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user 35872
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.
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Homotopy type of sphere
Let $X$ be a $CW$-complex of finite dimension.
Suppose that $\pi_k(S^n)$ is isomorphic to $\pi_k(X)$ for all $k\geq 0$, where $n>1$.
Certainly, $\dim X\geq n$. It is easy to show that $X$ has homo …
- 369
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Induced maps on homotopy groups by self maps of $\mathbb{CP}^n$
Let $f:\mathbb{S}^2\to \mathbb{S}^2$ with degree $d$.
It is well known that the induced map $$f_\ast:\pi_3(\mathbb{S}^2)=\mathbb{Z}\to \pi_3(\mathbb{S}^2)=\mathbb{Z}$$ is given by multiplication by $ …
- 369
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Path component of the mapping spaces $G(S^m, S^n)$
Let a non trivial $\alpha\in \pi_m(S^n)$ with a finite order $|\alpha|$. Write $G_\alpha(S^m, S^n)$ for the path-component determined by $\alpha$ of the mapping space $G(S^m, S^n)$ of free maps. Next …
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P.J. Hilton notes requested
Does anybody here have the mimeographed notes Homotopy theory and duality, by P.J. Hilton, Cornell University, 1959 ?
I guess that those notes were never published online.
I believe that some topol …