11,955 reputation
22566
bio website homepages.abdn.ac.uk/…
location Aberdeen, United Kingdom
age
visits member for 4 years, 9 months
seen 4 mins ago

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


2h
comment stable splitting into a wedge sum
What's the map from $X$ to its suspension?
18h
accepted Non-orientable $6$-manifold with $H_4(M)=\mathbb{Z}/2$?
1d
comment Non-orientable $6$-manifold with $H_4(M)=\mathbb{Z}/2$?
@user51223: This question came up when studying mod 2 cohomology classes realizable by immersions/embeddings.
1d
comment Non-orientable $6$-manifold with $H_4(M)=\mathbb{Z}/2$?
@GabrielC.Drummond-Cole: Thanks, I think it works!
2d
asked Non-orientable $6$-manifold with $H_4(M)=\mathbb{Z}/2$?
May
12
comment Triangulations of submanifolds of smooth manifolds
@IgorRivin: That's a good question, I confess to not being an expert. Here's what I know: $\operatorname{Emb}(M,N)$ is open in $C^\infty(M,N)$, and dense if $2m\le n$. Also, the space of topologically stable maps $M\to N$ is open and dense in $C^\infty(M,N)$ (the Thom-Mather theorem). I believe it follows that every embedding is isotopic to a topologically stable, hence triangulable, embedding.
May
11
awarded  Necromancer
May
10
answered What are the uses of the homotopy groups of spheres?
May
10
answered Triangulations of submanifolds of smooth manifolds
May
5
comment Reference request: Flipping the factors in the Künneth formula
@SeanTilson: I disagree. This is exactly the kind of thing you would not want to reprove in a paper, hence seems like a totally reasonable reference request.
Apr
21
comment Studying topology: which first, algebraic or differential?
I downvoted this otherwise excellent list, but I must admit it was an emotional reaction to the negative comments about Hatcher's book. Now the software won't allow me to reverse my downvote, unless the answer is edited. Perhaps this will motivate you to remove these subjective comments? ;)
Apr
21
comment Example s.t. the unbased loop-space is not $\Omega X \times X$
A convenient low-tech reference for the "standard fact" is in V. L. Hansen, On the fundamental group of a mapping space. An example, Compositio Mathematica 28, 1974.
Apr
7
comment Cohomology operators inducing local basis of $1-$forms
What do you mean by local basis here? If you just want a basis of $\Omega^1(M)$ of the given type, then it seems like $H^1(\Omega(M),\partial)=0$ is a necessary and sufficient condition.
Apr
3
comment Toral rank conjecture
@MyIsmail: I think the $2$ in the second bound should be $1/2$.
Mar
25
comment Homotopy of orthogonal groups in the unstable range
Section 1 of the paper maths.ed.ac.uk/~aar/papers/levsph.pdf contains a similar result for $SO$. Perhaps the argument there works for $O$ also?
Mar
19
comment cohomology of the orbit space of a group action
Presumably this can be promoted to a statement about graded algebras? And is it also true if $|G|$ is invertible in $F$?
Mar
17
comment Two H-space structures on S^3 and [X,S^3] different as groups for each: Explicit Example?
Nice! And you might hope to compute the groups $[S^3\times S^3,S^3]$, using the cofibration sequence $S^3\vee S^3\to S^3\times S^3 \to S^3\wedge S^3 = S^6$.
Mar
17
comment Two H-space structures on S^3 and [X,S^3] different as groups for each: Explicit Example?
If $X$ is a closed oriented $3$-manifold, then maps $X\to S^3$ are classified by their degree. Since the degree can be detected homologically, and since any multiplication $m:S^3\times S^3\to S^3$ must do the obvious thing on third homology, I think all the groups are isomorphic when $X$ is an oriented $3$-manifold. (I leave this comment in case anyone else was thinking of using Hopf's theorem.)
Feb
22
comment Pairing of cohomology and homology Künneth formulas
@user43326: Good point. Well, it's not trivial to me! But I would expect (if true) it should be proved somewhere already.
Feb
21
comment Pairing of cohomology and homology Künneth formulas
The pairing itself is canonical, but you are right that the way Tor sits in the homology depends on a choice of splitting. I think the statement could still be true, but would also accept counter-examples as answers!