# Questions tagged [at.algebraic-topology]

Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

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### Locally trivial fibration over a suspension

For $X$ a paracompact space, I am trying to classify all locally trivial fibration, with fiber-type a space $F$ such that $G_F = Homeo(F)$ with the C.O. topology is a topological group, and base the ...

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20 views

### homology of the fiber

Let $f:X\rightarrow Y $ be a fibration (with fiber $F$) between simply connected spaces such that
$H_{\ast}(f):H_{\ast}(X,\mathbb{Z})\rightarrow H_{\ast}(Y,\mathbb{Z})$ is an isomorphism for $\ast\...

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80 views

### $X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a Co-H-Space

I have asked the below question on MathSE (with a 200 point bounty) but have yet to receive an answer there, and so am trying here. I am happy to remove it if it is nevertheless decided that this ...

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172 views

### What is the total square on the dual Steenrod algebra?

The dual Steenrod algebra ($p=2$) has generators $\xi_n$ and these have conjugates that are often labeled $\zeta_n$. I am curious about the left and right actions of the Steenrod algebra on its dual, ...

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269 views

### When is a map of topological spaces homotopy equivalent to an algebraic map?

My question is simple, but I don't expect there are any simple answers.
Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points....

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### Higher homotopy groups of Calabi-Yaus

Is something known about the higher homotopy groups of Calabi-Yau threefolds? For example, one of the easiest CYs is the quintic, defined as the anticanonical divisor in $\mathbb{CP}_4$. What are its ...

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### Homotopy classes of maps between special unitary Lie group. Correction [duplicate]

An hour ago I asked a question (under the same title) but I used a wrong notation. Here is the improved version.
We consider a special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and ...

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71 views

### Action of the symmetric group on connected sums of manifolds (minus a disk)

Let $M$ be a connected compact topological $n$-dimensional manifold without a boundary and with a CW-structure $M= \bigcup M^i$. We have that
$$ (\#^g M)\smallsetminus D^n \simeq \bigvee_{i=1}^gM^{n-...

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185 views

### Homotopy classes of maps between special unitary Lie groups

I am sorry to mislead the notations: $SU(n)$ should be replaced by $PSU(n)$. I will reformulate it now.
We consider the special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and we ...

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### Examples of odd-dimensional manifolds that do not admit contact structure

I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?

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### Homotopy of group actions

Let $G$ be a topological group and $X$ be a topological space.
Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous ...

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294 views

### Counter-example to the existence of left Bousfield localization of combinatorial model category

Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ?
It is well known to exists ...

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### Generators of the fundamental group of the solid torus [migrated]

I have a solid torus $T$ and a curve (or knot) $C$ that winds 2-times around the torus (parallel to the longitude). Can I say that $[C] = 2 \in \mathbb{Z} \cong \pi_1(T)$, that is the curve represents ...

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### Is the Thomason model structure the optimal realization of Grothendieck's vision?

In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...

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202 views

### Topological data of $K3\times T^{2}$

I need some help in order to clarify some topological data of a $K3\times T^{2}$ Calabi Yau manifold in which $K3$ part has been obtained as a resolution of a $T^{4}/ \mathbb{Z_{2}}$ orbifold .
EDIT:...