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Results tagged with homotopy-theory
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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.
3
votes
Accepted
Connective covers of based spaces
Such a tower is usually called a Whitehead tower. More generally, there is the notion of a Moore-Postnikov factorization of a map $f: X \to Y$ of spaces: fixing an $n$, we can find a factorization $X …
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13
votes
Accepted
On the definition of stably almost complex manifold
The definition is, indeed, equivalent to a choice of complex structure on $TM \oplus \Bbb R^{k}$ for sufficiently large $k$ (roughly $k > dim(M)$). Roughly, this is because the tangent bundle and the …
- 52.7k
1
vote
Accepted
Homotopy coherent transformation and totalization
The answer is that the double complex does not collapse in the case of $HX$ either. The issue is that you have a coherent simplicial diagram valued in chain complexes with differential zero, rather th …
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8
votes
Accepted
Does a ring spectrum with even homotopy and even cells always have a polynomial algebra of h...
Let $G$ be any discrete group, and let $MU[G] = MU \otimes \Sigma^\infty_+ G$ be the associated group algebra over $MU$. Additively, $MU[G] \simeq \bigoplus_{g \in G} MU$, and so it has both even homo …
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12
votes
Accepted
Does the Atiyah-Hirzebruch spectral sequence for $E^\ast(X)$ collapse whenever $E$ is comple...
Yes, this is true.
As you commented, $X$ has finitely generated free homology, and so the $E_2$-term of the AHSS can be identified with
$$
H^*(X; E^*) \cong E^* \otimes H^*(X).
$$
To show collapse, we …
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3
votes
Accepted
homotopic to a constant map
Let $X = \Bbb{RP}^n$ for $n \geq 2$. I claim that the map $X \to X^3 / M$ is nontrivial on mod-2 cohomology.
Here is some general material.
For $1 \leq i \leq 3$, let $p_i: X^3 \to X^2$ be the projec …
- 52.7k
7
votes
Accepted
What is the group completion of finite sets with respect to cartesian product?
As already addressed in the comments:
Group completing the groupoid of finite pointed sets under the smash product gives a contractible space.
The groupoid of finite sets under the cartesian product …
- 52.7k
3
votes
Accepted
Bar construction in commutative algebras is calculated by pushout
This is just to be explicit about the role of the bar construction in David's answer.
If $I$ is the category $a \leftarrow b \rightarrow c$ parametrizing pushout diagrams, then there is a functor $f: …
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8
votes
Accepted
Does the forgetful functor $F:\mathrm{CAlg}\to\mathrm{Alg}^{(1)}$ sending $E_\infty$-ring sp...
Limit-preservation is the content of section 3.2.2 of Higher Algebra (see particularly Corollary 3.2.2.5). Preservation of sifted colimits is in section 3.2.3 (see particularly Corollary 3.2.3.2).
As …
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3
votes
Accepted
Symmetric-monoidal-associative smash product up to homotopy
We don't just have a problem with commutativity of the structure maps in the stable category; there are also not enough premaps to construct a map $\tau$ representing the twist automorphism in the sta …
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26
votes
What are some toy models for the stable homotopy groups of spheres?
My favorite warmup example to the stable homotopy groups of spheres is the following differential graded algebra.
Let $A$ have the underlying ring
$$
\Bbb Z[y] \otimes \Lambda[x],
$$
a ring with a po …
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9
votes
Accepted
How many automorphisms are there of the category of filtered spectra?
Here is a proof that any autoequivalence of (the $\infty$-category of) filtered spectra is naturally equivalent to a suspension functor. Since filtered spectra are equivalent to simplicial spectra, th …
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2
votes
Cut a homotopy in two via a "frontier"
This addresses the revised question.
There are examples where this is impossible. For example, define subsets of $[0,1]^2$ by
$$
\begin{align*}
A &= \{(x,y) \mid y > 0, x \leq \tfrac{1}{2} + \tfrac{\s …
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8
votes
Chromatic representation theory of the symmetric groups?
The coefficient ring of $K(h)[\Sigma_n]$ is, in degree zero, $\Bbb F_p[\Sigma_n]$, the group algebra on $\Sigma_n$ over $\Bbb F_p$. As a result, the list of idempotents in this ring is the same as for …
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7
votes
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The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus...
Z.M.'s first comment is correct. If $S \subset \Bbb Z \cong \pi_0(\Bbb S)$ is a multiplicatively closed subset, there is a localization $S^{-1} \Bbb S$ whose homotopy groups lift the algebraic localiz …
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