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location  Aberdeen, United Kingdom  
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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 nontrivial 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 nonabelian topological group $G$ for which ${\rm aut}_1(G)$ is homotopy abelian would give a counterexample (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 nullhomotopic. 
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 nullhomotopic 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. 