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Results tagged with homotopy-theory
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user 61328
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.
14
votes
2
answers
857
views
Eilenberg-Mac lane spaces and a generalization
Let $G$ and $H$ be two abelian groups and let $n>1, m>1$ be two different integers. How many different spaces $X$ (up to homotopy) do we have with the property $\pi_{n} X=G$ , $\pi_{m} X=H$ and $\pi_{ …
- 1,607
11
votes
2
answers
761
views
from a circle to higher spheres
Question: Is there a group $G$ and a CW-complex $X$ such that
1) $X$ is homotopy equivalent to the circle $S^{1}$.
2) $G$ acts on $X$
3) the space of fixed points $X^{G}$ is weakly equivalent to …
- 1,607
4
votes
1
answer
340
views
localization and $E_{\infty}$-spaces
Let $\mathrm{Top}$ be the model category of topological spaces. Define a new model structure on $\mathrm{Top}$ where $f:X\rightarrow Y$ is a weak equivalence iff $$f_{\ast}:H_{\ast}(X,\mathbb{F}_{p})\ …
- 1,607
8
votes
1
answer
446
views
Loop space generalization
Let $X$ be a based connected space. The space of based continuous morphisms $Top_{\ast}(S^1,X)$
is the space of loops $\Omega X$. Since $S^1$ is homotopy equivalent to the Eilenberg-Mac Lane Space $K …
- 1,607
3
votes
1
answer
1k
views
fixed point and homotopy fixed points
Let $G$ be a group and $X$ be a $G$-space (finite G-CW-complexe when needed).
Let $p$ a prime number and $G= \mathbf{Z}/p\mathbf{Z}$,
If I'm not wrong Miller-Lannes,... theory provides tools and cr …
- 1,607
5
votes
2
answers
783
views
$E_n$-space and n-connected pointed space
Is it true that the homotopy category of group-like $E_n$-spaces is equivalent to the homotopy category of pointed $n$-connected spaces ? If it is true, what should be the statement when $"n\rightarr …
- 1,607
6
votes
1
answer
1k
views
homotopy fixed points and fixed points
Let $X$ a smooth projective scheme over a field $k$. And let $THH(X)$ denotes the topological Hochschild homology of $X$. Recall that the spectra $THH(X)$ admits an action of the of circle $S^{1}$. L …
- 1,607