# Questions tagged [homotopy-theory]

Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in algebraic topology.

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### Relationship between homotopy pushout and ordinary pushout

I'm trying to understand the homotopy pushouts and currently looking at the homotopy cofiber. For two maps $f \colon C \to A$ and $g \colon C \to B$ we defined the homotopy pushout to be the regular ...
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### Compact space homotopy equivalent to a CW complex

Assume that $X$ is a compact Hausdorff space homotopy equivalent to some CW complex. Does it follow that it is homotopy equivalent to a compact CW complex?
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### Is there an operad whose algebras are homotopy commutative $E_1$-algebras?

I might guess that the Boardman-Vogt tensor product of the $E_1$ operad and the $A_2$ operad might do the trick. That is, I would guess that an $A_2$ object in $E_1$ algebras, or equivalently an $E_1$ ...
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### Intuition behind the injectivity part of Hurewicz Theorem

The surjectivity part of Hurewicz Theorem is easy to understand: under the inductive hypothesis that all homotopy groups (of a CW-complex) up to dimension $n$ are trivial, it is clear (I believe) how ...
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### Taking homotopy fixed points preserves fibrations

I'm reading a paper where they have an appendix about homotopy fixed point sets of a $G$-space, and at some point they claim that if $f:X\to Y$ is a $G$-map that is an ordinary (non-equivariant) ...
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### Non-existence of pushout in homotopy category

I want to show that $S^1_{(0)}\leftarrow *\to S^1_{(1)}$ has no pushout in the homotopy category without using Eilenberg–MacLane spaces. In a first step, I want to show that if there is such a ...
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