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bio website math.cornell.edu/~hatcher
location Cornell University
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visits member for 4 years, 8 months
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May
17
revised Intuition behind Thom class
corrected a typo in a formula
Feb
19
awarded  Enlightened
Feb
19
comment Seifert fiberable manifolds with several Seifert fiberings
@Werner Thumann: That is correct, each prism manifold has exactly two Seifert fiberings. This is an interesting contrast to lens spaces, $S^3$, and $S^1\times S^2$, each of which has infinitely many different Seifert fiberings.
Feb
18
awarded  Nice Answer
Feb
16
revised Seifert fiberable manifolds with several Seifert fiberings
added 45 characters in body
Feb
16
answered Seifert fiberable manifolds with several Seifert fiberings
Feb
1
comment Adams' theorems on the Hopf-Whitehead J-homomorphism
For question 1, I think there is a similar sort of argument to show injectivity in these cases, though it needs a bit more input, notably Adams operations in real K-theory. I have some handwritten notes on this from 20 years ago that would take some work to decipher after this long a time. My recollection is that I extracted these from Adams' J(X) - IV paper. My plan was, and still is, to include this in that unfinished book mentioned in the question, though as the years pass the chances of this ever happening become increasingly slim.
Jan
21
comment Foliation of surface all of whose leaves are circles
One can also reduce the case of nonempty boundary to the closed case by doubling: Take two copies of the surface with boundary and identify their boundaries to get a closed surface. If you know the closed surface is a torus, the original surface must then be an annulus. (Doubling doubles the Euler characteristic.)
Jan
20
answered Moore decomposition, dual to Postnikov tower
Dec
26
answered Action of Mapping Class Group on Arc complex
Dec
16
comment Codimension zero embeddings and diffeomorphism groups
The paper of LaBach mentioned in an earlier comment (now deleted?) uses a nonstandard topology on $Diff(D^n)$. The restriction map $Diff(D^n)\to Diff(int(D^n))$ is injective so can be viewed as an inclusion, and LaBach uses the subspace topology on $Diff(D^n)$ induced from the compact-open topology on $Diff(int(D^n))$. In this topology one can do a sort of "reverse Alexander trick" and push all the complications of a diffeomorphism of $D^n$ out to $\partial D^n$ and make them disappear.
Dec
16
comment Codimension zero embeddings and diffeomorphism groups
$Diff(D^n)$ is homotopy equivalent to $O(n)\times Diff(D^n\ rel\ D^{n-1})$ where $D^{n-1}$ is a disk in $\partial D^n$. The factor $Diff(D^n\ rel\ D^{n-1})$ can be identified with the pseudoisotopy space $Diff(D^{n-1}\times I\ rel\ D^{n-1} \times 0)$, so it has a complicated homotopy type when $n$ is large enough.
Dec
16
awarded  Enlightened
Dec
15
awarded  Nice Answer
Dec
14
revised Homotopy versus path-homotopy on punctured surface
more precise statements
Dec
14
answered Homotopy versus path-homotopy on punctured surface
Dec
14
awarded  Nice Answer
Dec
13
awarded  Necromancer
Dec
13
answered Triangulating surfaces
Dec
13
comment Triangulating surfaces
It should be pointed out that the Gallier-Xu book does not give a complete proof of triangulability of surfaces, contrary to what seems to be claimed in the preface to the book. The hardest step in the proof of Thomassen that they present in Appendix E is the Schoenflies theorem, and they only state this without including a proof. To make matters worse, immediately after the statement of the Schoenflies theorem they make the erroneous claim that it can be proved using tools of algebraic topology, but this is true only for the weaker Jordan curve theorem.