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For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in su …

Let $F$ be a finite group. Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially? For example the Bar construction has the proper …

I looked at my homotopy theory lecture notes and we had the following similar result: $X$ H-CoGroup, $Y$ $H$-Group, then both group structures defined on [X,Y] agree. The proof goes roughly as follows …

Let $X$ be a space and $I$ be an ideal in the cohomology ring $H^*(X)$. I am interested in the question whether there is a map of spaces $Y\rightarrow X$ such that $I$ is the kernel of the induced map …

Wikipedia already gives a list of some examples.

Suppose we have a cobordism $W$ of manifolds $M_0$ and $M_1$ and suppose the inclusion of $M_0$ into $W$ is a homotopy equivalence. Is the same true for the inclusion of $M_1$ (ie. is $W$ already an h …

View $B^{n}$ as $(S^{n-1}\times[0,1])/(S^{n-1}\times \{1\})$. This means that an extension of a map from $B^{n}$ to somewhere is just an extension to $S^{n-1}\times [0,1]$ which is constant on the oth …

I was wondering how higher cohomology operations interact with spectral sequences. For example if $X$ is a simplicial set. Then $C^*(X,\mathbb{F}_2)$ has an algebra structure over the Surjection Opera …

Suppose $S^1$ acts on $S^2$ by rotation around one axis. This action commutes with the antipodal map and hence gives an action on $\mathbb{R}P^2$. But $\mathbb{R}P^2/S^1\cong [0,1]$ and hence this can …

For simplicity I want to stick to the compact, orientable case. Further i want to assume, that $N$ has the structure of a CW-complex. It might still be interesting to consider the case, where the dim …

Since I am most familiar with the (co)chain complex version (using the surjection operad (cf. McClure-Smith)), I will give an example in this setting. The surjection operad $S$ is also filtered by the …

I will make use of the surjection operad to produce an explicit example of a (functorial) cochain whose boundary is the Adem-Relation. Let us show that $Sq^1Sq^1([x])=0$ for a cohomology class $x$ in …

Definition: The complexity of a surjection $\{1,\ldots,n+k\}\rightarrow \{1,\ldots,n\}$ is defined in the following way. First think of this map as the tuple $(f(1),\ldots,f(n+k))$. For two numbers …

I will call the property of a $G$-CW-complex that inside every neighborhood of a point one can find a $G$-invariant neighborhood property $A$. As in your example, a graph where an edge stabilizer has …

Here are some examples. And since this really just adresses the computation part of the question and not what that map is, it is really just a partial answer. LEt me first describe the sequence oper …