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Lecturer at University of Aberdeen, with interests in Algebraic and Differential Topology and their applications.


1d
answered Unstable homotopy groups of spheres beyond Toda's range
1d
comment Unstable homotopy groups of spheres beyond Toda's range
Thanks, Ryan. This seems fairly definitive. I'll take a look at these references, and if nobody tells me that the results have been handily tabulated I'll accept this answer.
2d
awarded  Nice Question
2d
comment Homotopical nilpotency of self homotopy equivalence
@MyIsmail: No, that Proposition in the Salvatore paper says precisely what I said 3 comments above. The problem is that $\operatorname{nil}(G)$ may be greater than $\operatorname{Hnil}(G)$.
2d
asked Unstable homotopy groups of spheres beyond Toda's range
Dec
16
comment Schur covering group
I think it can't be true for a finite simple group with non-trivial Schur multiplier, eg, $A_n$ for $n>4$. I don't know about a general condition though.
Dec
12
comment Homotopical nilpotency of self homotopy equivalence
A few comments: The argument in the paper you linked to shows that ${\rm Hnil}(G)\le {\rm Hnil}({\rm aut}_1(G))$, basically because one is a retract of the other. I think it is unlikely that your stronger statement holds. Any non-abelian topological group $G$ for which ${\rm aut}_1(G)$ is homotopy abelian would give a counter-example (but I can't see if such things exist at present).
Dec
12
comment Homotopical nilpotency of self homotopy equivalence
@FernandoMuro: I don't understand your comment. The condition is about maps $(\operatorname{aut}_1(G))^{n+1}\to \operatorname{aut}_1(G)$, which surely can be null-homotopic.
Dec
8
comment Topological retraction vs categorical retraction
@MarcHoyois: I think Andrej's reading of the OP's definition of topological retract is "any space which is homeomorphic to a retract", and it seems that is what the OP intended. The fact that $A$ is given as a subset of $X$ is something of a red herring.
Dec
8
comment Topological retraction vs categorical retraction
@AndrejBauer: Yes, I really meant a very small circle which is obviously not a retract of the torus. Thus it does not satisfy (my interpretation of) condition 1. However, taken as a topological space (disregarding the embedding) it is clearly a categorical retraction, using the maps you give in your above comment.
Dec
8
comment Topological retraction vs categorical retraction
@Emil: It is not a retract, in the sense that there does not exist a map $r: X\to A$ which restricts to the identity on $A$. This can be seen by looking at first homology groups.
Dec
8
comment Topological retraction vs categorical retraction
@dominiczypen: Do you view $A$ as a topological space in its own right, or as a subset of $X$?
Dec
8
comment Topological retraction vs categorical retraction
@Andrej: In my reading of the question, $A$ is regarded as a subset of $X$ in condition 1, but as an abstract topological space (with the subspace topology inherited from $X$) in condition 2. In fact I was answering the original question (before your edit), but I still don't see what's wrong with my answer.
Dec
8
revised Topological retraction vs categorical retraction
added 66 characters in body
Dec
8
answered Topological retraction vs categorical retraction
Dec
8
comment Topological retraction vs categorical retraction
As Qiaochu says, this is a strange question, since the inclusion $A\subseteq X$ is manifest in the first condition but not the second. With the more generous reading, though, an example is given by any null-homotopic embedded circle in the torus.
Dec
5
asked When is the diagonal inclusion a $\Sigma_2$-cofibration?
Nov
28
awarded  Nice Question
Nov
20
revised Configuration space like subspace of sphere product
added 2 characters in body
Nov
20
comment Configuration space like subspace of sphere product
I think the algebraic map actually goes to $\mathbb{R}^{n+1}$, hence the discrepancy in our dimensions.