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Results tagged with homotopy-theory
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user 134512
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.
4
votes
Accepted
$\pi_1$ action on relative homotopy groups $\pi_n(X,A)$
I'd say the only tricky part is showing the action is well-defined, but if you trust it is here is a proof following Hatcher's proof for the basepointed version:
Let's consider the general case of ma …
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6
votes
Accepted
Decomposable maps of half-smash products
No, take the case $X=\ast$. This asserts that any map $S^n \rightarrow Z \rtimes Y$ factors through the inclusion of a fiber $Z \rtimes \{y\}$, up to homotopy. This is certainly false, for instance if …
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1
vote
Accepted
Multi-simplicial generalization of $\Gamma$-spaces
The paper "Iterated monoidal categories" by Balteanu, Fiedorowicz, Schwanzl, Vogt gives an analogous result for functors $(\Delta ^{op})^n \rightarrow Top$ and n-fold loop spaces.
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7
votes
Accepted
Can a restriction of a null-homotopic spherical map be null-homopotic?
There exists nullhomotopic maps preserving codimension one equators such that the restriction to the equators is not nullhomotopic. Very easy examples come from taking non nullhomotopic maps of sphere …
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5
votes
Turning injection of homotopy groups to an isomorphism
Consider $i: S^1 \hookrightarrow M_f$ where $f:S^1 \rightarrow S^1$ is squaring and $M_f$ denotes the mapping cylinder. Then the inclusion induces a map $\pi_1(S^1) \rightarrow \pi_1(M_f)$ which has i …
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3
votes
Can one define relative Hurewicz maps using the Dold-Thom theorem
In the absolute case, $\pi_*(X) \rightarrow \pi_*(SP(X))$ can be identified with the Hurewicz map because one can explicitly compute $\mathbb{Z}=\pi_n(S^n) \rightarrow \pi_n(SP(S^n))=\mathbb{Z}$ and s …
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1
vote
Accepted
Preservation of fiberwise normal bundles under fiberwise homotopy equivalences
After looking at the actual construction of the fiberwise embedding of a bundle $M \rightarrow E \rightarrow B$ into $B \times \mathbb{R}^N$, it became clear how to adjust it to embed the mapping cyli …
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3
votes
Accepted
[M,N]≅ [M,R] ⊗ N for E-infinity modules
See page 70 of EKMM. The map $$F_R(M,N) \wedge_R F_R(M',N') \rightarrow F_R(M \wedge_R M', N \wedge_R N')
$$ is an isomorphism if $M,M'$ are strongly dualizable or if $M$ is strongly dualizable and $N …
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9
votes
Accepted
Can a simply connected manifold satisfy $𝑀\simeq 𝑀\times 𝑀$?
Thanks to Dave Benson for pointing out an algebra error in the first draft of this answer and a missing detail.
Suppose $X$ is homotopy equivalent to a finite dimensional, simply-connected, and noncon …
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13
votes
$\infty$-categorical description of $n$-manifolds
There is a specific "$\infty$-categorical" approximation of manifolds which your question seems to converge to. Before we address that, I would like to point out that Ayala-Francis do not classify exc …
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2
votes
Accepted
Cofibrancy of a right module over an operad
If $O$ is a reduced operad in $(C,\otimes)$, i.e. one with $O(\{*\})=1$ and $O(\emptyset)=*$, then under mild assumption on $(C,\otimes)$, the projective model structure exists on $\mathrm{RMod}_O$, i …
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31
votes
Accepted
If homotopy groups of spaces are identical, then stable ones are also identical?
No, a counterexample is the rational sphere $S^{2n}_\mathbb{Q}$ and $K(\mathbb{Q},2n) \times K(\mathbb{Q},4n-1)$. By the work of Serre these have the same homotopy groups, though it is easy to see the …
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0
votes
Does the bordism homology theory satisfy the weak equivalence axiom?
Thom directly constructed a natural isomorphism $MO_n(X) \rightarrow \pi_n (MO \wedge X_+)$ with no assumptions on $X$. The latter sends weak equivalences to isomorphisms since it is computed as a col …
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3
votes
Find a functorial zig-zag of spaces
If the diagrams $X_\bullet$ and $Y_\bullet$ are constructed canonically, but not necessarily naturally, from spaces $X,Y$ where $X \simeq Y$, one technique to construct such a zigzag is pick $f: X \si …
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4
votes
Accepted
Delooping a weak $E_1$ map by bar construction
No, take $Z$ a connected space for which $\Omega Z$ is homotopy commutative but $Z$ has no $A_\infty$ multiplication, e.g. $Z$ could be an $H$-space which doesn't have the homotopy type of a loop spac …
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