Tom Goodwillie

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Name	Tom Goodwillie
Member for	3 years
Seen	1 hour ago
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Location	Brown University
Age	58

algebraic (?) topologist, big fan of the empty set

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2d	comment	Tangent space to positive oriented Grassmannians Why should positivity matter?
Jun 16	comment	What is the algebraic geometry version of the spheres? In homogeneous coordinates $((a:b),(c:d))\mapsto (ac:ad+bc:cd)$. More conceptually, multiply homogeneous linear polynomials $(aX+bY)(cX+dY)$. $P^n$ is the $n$-th symmetric power of $P^1$, by unique factorization of degree $n$ homogeneous polynomials.
Jun 9	comment	Embedding tower in low codimension I think you can argue that the limit of the tower for knots has uncountably many components--the map from $T_k$ to $T_{k-1}$ is surjective but not injective on $\pi_0$.
Jun 8	awarded	● Yearling
Jun 7	comment	A category with weak equivalences which is not a model category Do you really want to call it a "category with weak equivalences" if not every identity map is a weak equivalence?
Jun 7	comment	A category with weak equivalences which is not a model category @Karol: See my comment above. Any such model structure would be such that every trivial fibration is an isomorphism, therefore every map is a cofibration.
Jun 7	comment	One question on cup product and torsion elements Oh, now I see that Allen mentions this as a special case of his example.
Jun 7	comment	A category with weak equivalences which is not a model category If the class of weak equivalences is as Peter says, or smaller, then every trivial fibration $X\to Y$ is an isomorphism. Proof by induction that $X_n\to Y_n$ is bijective: Given $y:\Delta^n\to Y$, pull back the map to get a trivial fibration $y^\ast X\to \Delta^n$. An object mapped to $\Delta^n$ is at most $n$-dimensional (this is really the crux), and if the map to $\Delta^n$ is an isomorphism on the $(n-1)$-skeleton then it cannot be a weak equivalence without being an isomorphism.
Jun 6	answered	One question on cup product and torsion elements
Jun 5	awarded	● Enlightened
Jun 5	accepted	Integral cohomology (stable) operations
Jun 4	comment	sheaf of differential forms - tangent sheaf [Hartshorne] For smooth manifolds there are several ways of thinking about (co)tangent spaces and bundles and sheaves. Some, but not all, of these ways of thinking can easily be adapted to algebraic geometry. To get the idea, I suggest thinking about what the definitions that you are reading about in Hartshorne would give if you applied them to $C^\infty$ functions on a manifold.
May 31	awarded	● Nice Answer
May 30	comment	homotopy pullback/pushout Dylan, in some sense it's more elementary than Blakers-Massey. To get the idea, here's why $X\to \Omega\Sigma X$ is a weak equivalence when $X$ is a spectrum: $X$ consists of spaces $X_n$; the map is given by Freudenthal maps $X_n\to \Omega\Sigma X_n$; an inverse on spectrum homotopy groups is given by the obvious maps $\Omega\Sigma X_n\to \Omega X_{n+1}$.
May 26	comment	Inverse Function Theorem in Algebraic Geometry In some trivial but profound sense every etale map is locally an isomorphism. Let "locally" mean "locally in the etale topology".
May 26	comment	Quotient of 3-sphere by binary octahedral group? This 'octahedral group' is isomorphic to the symmetric group $S_4$. Its abelianization has order two, and the abelianization of the 'binary octahedral group' is the same.
May 25	comment	Quotient of 3-sphere by binary octahedral group? What is it that you know about the Poincare sphere but don't know about this other manifold? It's homology? A description as a polytope with sides identified?
May 24	awarded	● Good Answer
May 24	awarded	● Necromancer
May 23	revised	G-equivariant Whitehead’s Theorem added 16 characters in body
May 23	accepted	G-equivariant Whitehead’s Theorem
May 23	answered	G-equivariant Whitehead’s Theorem
May 21	comment	Chern Character Isomorphism for non-finite CW complexes, resp. for non-CW complexes For a compact Hausdorff space $X$ you can describe both Cech cohomology and K-theory as direct limits over finite complexes that $X$ maps to. For locally compact Hausdorff spaces you might want to consider the one-pojnt compactification.
May 17	comment	Waldhausen $K$-theory for $G$-spaces Thanks, Dylan. If none of them can help me, I'll ask you about non-humans.
May 16	comment	Cyclotomic fields Ah, now I see. Forgive my flippant comment. I am fond of the number zero.
May 16	comment	Cyclotomic fields I suspect that Emil made the mistake of looking at $\zeta_{p^0}-1$.
May 16	asked	Waldhausen $K$-theory for $G$-spaces
May 9	comment	Homotopy equivalences preserving structure When you say "for each nonempty intersection", it's not immediately clear whether by "nonempty" you mean that the intersection of the $X_{i_j}$ is nonempty or that $r>0$. But clearly (looking at your base case) you cannot mean the latter. And if the former then you seem to be forgetting that an empty intersection of $X_i$s might map to a nonempty intersection of the corresponding $Y_i$s. Why say "nonempty" at all?
May 5	comment	Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field? This is more elementary than the Whitney trick. Two zeroes of opposite index can be taken to lie in a coordinate chart (because the manifold is connected) and then they can be eliminated using that fact that a map $S^{n-1}\to S^{n-1}$ of degree zero extends to $D^n$.
Apr 24	comment	Can Inequivalent Topologies Have Same Sheaves/Cohomology? I wouldn't expect anything involving sieves to hold water, but I think this does.
Apr 23	comment	A Property of Finite Rings Let $C$ be the set of all ordered pairs $(x,y)\in R\times R$ such that $xy=0$. Then $\|C\|$ is the sum, over all $x\in R$, of $\|A(x)\|$. On the other hand $C$ is also the sum, over all $y\in R$, of $\|B(y)\|$.
Apr 23	comment	A Property of Finite Rings I gave the reason.
Apr 22	accepted	A Property of Finite Rings
Apr 14	comment	When does the sheaf cohomology of a topological space vanish? (The Cantor set is homeomorphic to $\mathbb Z_p$.)
Apr 14	comment	Analogy between the exterior power and the power set I wasn't really thinking of a quadratic form. I was just thinking that, like a Clifford algebra, the power set has a filtration such that the associated graded object is (like) an exterior algebra.
Apr 13	comment	When does the sheaf cohomology of a topological space vanish? If every open cover of $X$ has a refinement consisting of disjoint open sets, then every (locally) surjective map of sheaves on $X$ is surjective on global sections, so that higher sheaf cohomology is trivial.
Apr 13	comment	Analogy between the exterior power and the power set Maybe the power set is more like a Clifford algebra.
Apr 11	comment	Asymptotics of the Number of Non-Isomorphic Equivalence Relations and the Number of Non-Isomorphic Relations OK, I'm considering it.
Apr 10	comment	Asymptotics of the Number of Non-Isomorphic Equivalence Relations and the Number of Non-Isomorphic Relations It appears to me that for every equivalence relation $E$ there are at least $n$ nonisomorphic ways of making a relation $R$ such that $E$ is generated by $R$. If so, then $\frac{p(n)}{a(n)}\le \frac{1}{n}$.
Apr 10	comment	Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$ That step is wrong. I'll fix the proof.
Apr 10	comment	When is Ad(pi) an irreducible representation ? There are examples where $\pi$ is self-dual and $n=2$.
Apr 10	revised	Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$ added 287 characters in body
Apr 9	comment	Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$ Good point. I'm sure that I never in my life thought that every matrix over an algebraically closed field is diagonalizable, but I did mess up somehow. I've corrected some typos, and will try to corret the thinko soon.
Apr 9	revised	Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$ edited body
Apr 8	accepted	Coefficients of real k-theory with coefficients
Apr 5	answered	Coefficients of real k-theory with coefficients
Apr 3	accepted	Sum of two tangent bundles of $S^{2n}$
Mar 31	revised	Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$ deleted 1 characters in body
Mar 31	answered	Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$
Mar 30	awarded	● Good Answer

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