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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 …

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 …

Given a quasiisomorphism of DGAs $f:A\rightarrow B$ and a DG-module $M$ over $A$. Is the canonical chain map \[M\rightarrow B\otimes_A M \qquad m\mapsto 1\otimes m\] an isomorphism on homology?

This is an elaboration on the L's comment. Let me use the cochain convention, so that the differential has degree +1. Let $B=\mathbb{Z}/2$ (concentrated in degree 0) and let $A = \mathbb{Z}[e]/(e^2,d …

Let me just ignore the appearing decorations. I think the argument should work with all decorations. The $L$-groups of an additive category $\mathcal{A}$ with involution are defined as the cobordism c …

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 …

I do not know, in which category your actions live. In the following answer I want to consider isometric actions. Another classification problem might be: Classify all periodic one parameter subgrou …

You might want to have a look at the paper "Orientation reversal of manifolds" by Daniel Muellner.

Let $X$ be a (finite) simplicial complex and let $f$ be a map from the set of its $n$-Simplices to a abelian group $A$, with the property, that every cycle maps to $0$ (extending $f$ linearly). Let $ …

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 …

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 …

A nice example is the following. Let $G=\mathbb{Z}/p$ and look at the DGA $C^*$ of singular cochains on $BG$ with coefficients in $\mathbb{F}_p$. Let us assume $p>2$. Its cohomology is an exterior al …

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 …

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 …

Here is a counterexample, which is probably not "nice". Let $X$ be the Warsaw-circle. Let $X_n$ be the obtained from the Warsaw-circle by thickening the limit inverval by $1/n$. The intersection of al …