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# Questions tagged [at.algebraic-topology]

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

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### simplicial nomenclature and homology

Suppose I have a simplicial complex $K$ constructed by taking two simplicial complexes $K_1$ and $K_2,$ and coning off ever vertex of $K_1$ to all of $K_2$ and vice versa (so, a direct generalization ...
1answer
75 views

### Group completion of $E_k$-algebras

Let $X$ be an $E_k$-algebra. We can form the delooping $BX$, which is a $E_{k-1}$-algebra. The space $\Omega B X$ is again an $E_k$-algebra, which is grouplike (i. e. $\pi_0(\Omega B X)=\pi_1(B X)$ is ...
0answers
72 views

### A sufficient condition for $\pi_1(\operatorname B\Gamma) = 0$

I am currently reading Ghys and Sergiescu's paper Sur un groupe remarquable de difféomorphisms du cercle (French only I'm afraid), but part of their proof of Corollary 3.4 (page 20 of the pdf) is ...
1answer
166 views

0answers
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### Weak homotopy equivalence of sites

There are several notions of weak homotopy equivalence for topological spaces. The standard one can be formulated as follows: a map of spaces $X\to Y$ is a homotopy equivalence if the map of ...
0answers
62 views

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

We are given a free action of an abelian finite group on $S^3$. Let $L$ denote the quotient space and let an element $\alpha \in \pi_1 L =G$ be given. Does there exist a cyclic homotopy $h_t:L \to L$ ...
1answer
225 views

### Is there a Morita cocycle for the mapping class group Mod(g,n) when n > 1?

Write Mod(g,n) for the mapping class group of a genus-$g$ surface $\Sigma$ with $n$ boundary components. When $n=0,1$ we define the Torelli group $T$ to be the subgroup of Mod(g,n) which acts ...
0answers
98 views

### The model category structure on $\mathbf{TMon}$

I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer. I am trying to understand the homotopy colimit of a diagram of topological monoids, and ...
1answer
260 views

### 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?

I previously asked In which topological spaces does the existence of a loop not contractable to a point imply there is a non-contractable simple loop also? Given the broad scope of this question I ...
1answer
159 views

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

In another MathOverflow post I asked: 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? Note that ...
1answer
135 views

### Proving a Kan-like condition for functors to model categories?

I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...
3answers
2k views

### 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?

I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces. If it is true that: In a Topological Space, if there exists a loop that cannot ...
0answers
168 views

### Extending $K$-theory classes

Let $X$ be a $G$-space, for a compact Lie group $G$. If $U\subset X$ is a $G$-invariant open subspace, is it true that the restriction map on equivariant $K$-theory $$K_G(X)\rightarrow K_G(U)$$ is ...
1answer
183 views

### Quotient of arbitrary free involution on $S^n$

If we consider arbitrary free involution on $S^n$, then the quotient need not be diffeomorphic to $\Bbb RP^n$ if $n\geq 5$ and a reference for this is "some curious involutions of spheres" by Morris W....

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