# 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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### 2d TQFTs with values in simplicial sets and Reedy categories

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### What is the inverse in K-theory represented by Clifford module extensions?

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### Étale homotopy type of $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_\infty \}$

**2**

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### Filling condition for quasi-categories

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### What is the cotangent complex good for?

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### Products of cones and cones of joins

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### Homotopical characterization of CW complexes

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### Homotopical characterization of manifolds

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### Fundamental group and countability

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### Recovering operad units from homotopy units

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### Complement of contractible locally Euclidean subspace

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### Homotopy groups of ball complement

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### Every Spectral Deligne-Mumford stack satsifies fpqc descent?

**4**

**1**answer

### Different ways to “deloop” a (topological) $A_\infty$-algebra

**3**

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### Covers of a 4-manifold pull back a cohomology class to any algebraic multiple

**6**

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### Smooth covers pulling back a cohomology class to any algebraic multiple

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### Smooth complex projective surface as the total space of a Serre fibration

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### It is possible that $ X \simeq ΩX $? and that $ X \simeq Ω^ 2X $?

**3**

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### $(B\mathbb Z/p^{\infty})^{\wedge}_p\rightarrow (BS^1)^{\wedge}_p$ induced by inclusion is an equivalence

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### Classifying spaces of amalgamated topological monoids

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### Borel conjecture and arbitrary surface

**9**

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### Milnor excision for algebraic stacks

**32**

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### Equivalence of topological Hochschild homology and Mac Lane homology via an equivalence $QA\simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$

**5**

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### Contractibility of the category of cosimplicial resolutions

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### Is an (n-1)-sphere quotient by an (n-1)-sphere contractible? [closed]

**12**

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### Does every map $K(\mathbb{Z}, n) \to K(\mathbb{Z}/m, n + k)$ factor through $K(\mathbb{Z}/m, n)$?

**7**

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### A question on recognition of equivariant loop spaces

**3**

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### Étale homotopy equivalent varieties are deformation equivalent

**7**

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### Non-isomorphic complex Lie groups with the same exceptional Lie algebra for $\mathfrak{g_2,f_4,e_6,e_7,e_8}$?

**7**

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### When does a triangulated category have a heart?

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### Is there an exponential map in $\mathbb A^1$ homotopy theory?

**2**

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### On the link between homology and homotopy

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### Embedded ribbons and regular isotopy

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### Hopf invariants of elements from spherical fibrations

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### Any continuous map is homotopic to one assuming fixed values at finitely many points

**4**

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### Free abelian group on a space and fibrations

**4**

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### Interlocking (weak) factorization systems

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### When every closed and connected subset is path connected

**3**

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### About Homotopy Transfer Lemma

**4**

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### Homotopy descent and cohomology

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### One periodic cohomology theories?

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### Module spectrum maps up to stable homotopy

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### Intereresting classes of topological spaces locally modelled on some fixed spaces

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### Intuition for categorical fibrations?

**6**

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### (Pro-)representable functors and full subcategories in homotopy theory

**9**

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### If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic

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### Every immersion can be deformed to have only transverse self-intersections

**1**

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### Étale homotopy type of (derived) loop space

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### Essential Image of the Étale Homotopy type

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**1**answer