10,823 reputation
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bio website math.cornell.edu/~hatcher
location Cornell University
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visits member for 5 years, 5 months
seen 6 hours ago

Apr
5
awarded  gt.geometric-topology
Apr
4
answered Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
Mar
31
revised How can i change 8_19 to (3,4)-torus knot K(3,4)?
added 279 characters in body
Mar
31
answered How can i change 8_19 to (3,4)-torus knot K(3,4)?
Feb
24
comment Maps of balls with fixed value along boundary
On the other hand, if you wish to allow only homotopies $f_t$ and $g_t$ such that $f_t=g_t$ on $S^2$ for all $t$, then such pairs $(f,g)$ are equivalent to maps $h:S^3\to X$. Homotopy classes of maps $h:S^3\to X$ without basepoint conditions are in one-to-one correspondence with orbits of the action of $\pi_1(X)$ on $\pi_3(X)$, and this set of orbits can be nontrivial.
Feb
24
comment Maps of balls with fixed value along boundary
If you do not require $f=g$ on $S^2$ during homotopies, then the problem becomes trivial since every map $B^3\to X$ is homotopic to a constant map. Since we may assume $X$ is path-connected, this means $f$ and $g$ are homotopic.
Feb
23
answered Maps of balls with fixed value along boundary
Feb
16
awarded  Nice Answer
Feb
15
answered Homology equivalence and isomorphism on $\pi_1$ not enough for homotopy equivalence?
Feb
11
comment partial converse of existence of covering spaces
I've just posted an answer on the Stackexchange version of this question.
Feb
11
comment Inserting an open and simply-connected set between a compact set and an open set
Interesting use of language here (twice) that I don't recall seeing before: "If [X], if [Y], then [Z]." Is this equivalent to "If [X] and [Y], then [Z]"? Or maybe "If [X] such that [Y], then [Z]".
Feb
9
awarded  Guru
Feb
6
awarded  Good Answer
Feb
5
awarded  Enlightened
Feb
5
awarded  Nice Answer
Feb
5
answered Maps which induce the same homomorphism on homotopy and homology groups are homotopic
Feb
5
awarded  Good Answer
Dec
17
comment Cap product à la Poincaré
Also, $T^*$ is a cellulation (a decomposition into cells), not a triangulation as stated in the question.
Dec
17
comment Cap product à la Poincaré
You are talking about a product that involves just homology and not cohomology, so this is not the cap product. Instead it is usually called the intersection product. A classical reference for the intersection product is the textbook by Seifert and Threlfall. A more recent textbook source is Bredon's "Topology and Geometry". Bredon attributes the intersection product to Lefschetz. The intersection product is a perfectly well-behaved product, with the right hypotheses, so it's not clear why you use the words "evil", "fails", and "ill-defined" in reference to it.
Dec
9
answered Lens spaces and generalized Petersen graphs