# Questions tagged [homotopy-theory]

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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### Group completion of $E_k$-algebras

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### Are the real and complex Adams operations compatible under the inclusions $U(n) \rightarrow SO(2n)$?

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### Weak homotopy equivalence of sites

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### Cyclic homotopies of quotients of $S^3$

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### Homotopy pullback of motivic weak equivalences

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### In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point?

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### In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?

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### In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

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### Why does every chain complex have a map into its cone?

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### Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

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### Stable splitting of products

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### Equivalent definitions of Thom spectra

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### $L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles

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### Diffeomorphism type of the added sphere in simply connected surgery

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