User sam isaacson - MathOverflow

The semidihedral group of order 16 and ko

2012-02-23T20:17:20Z

Let $\mathcal{A}(1)$ denote the subalgebra of the $\mathrm{mod}\ 2$ Steenrod algebra generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$. The cohomology with $\mathbf{F}_2$ coefficients of the semidihedral group $$SD_{16} = \langle g, h \mid g^8 = h^2 = 1, hgh = g^3\rangle$$ of order $16$ is isomorphic to $\mathrm{Ext}^\ast_{\mathcal{A}(1)}(\mathbf{F}_2, \mathbf{F}_2)$. Is there a topological explanation for this isomorphism?

Answer by Sam Isaacson for Examples for non-naturality of universal coefficients theorem

2011-03-29T16:50:57Z

To expand on my comment, let $M = \mathbf{R}P^2$ be the Moore space with (reduced) homology concentrated in dimension $1$. Let $f:M \to \Sigma M$ be the map $$ M \to S^2 \to \Sigma M $$ given by collapsing the $1$-skeleton of $M$ and then including the bottom cell into $\Sigma M$. This map induces $0$ on $\tilde H_\ast({-};\mathbf{Z})$ for dimension reasons. However, the map $f$ is an isomorphism on $H_2({-};\mathbf{Z}/2)$; this follows from the long exact sequences $$ \dotsb \to H_2(S^1;\mathbf{Z}/2) \to H_2(M;\mathbf{Z}/2) \to H_2(S^2;\mathbf{Z}/2) \to \dotsb $$ and $$ \dotsb \to H_2(S^2;\mathbf{Z}/2) \to H_2(\Sigma M;\mathbf{Z}/2) \to H_2(S^3;\mathbf{Z}/2) \to \dotsb .$$ This is an example of "non-naturality."

To obtain a self-map of a space $X$ that is the identity on $H_\ast(X;\mathbf{Z})$ but not on $H_\ast(X;\mathbf{Z}/2)$, we follow Tyler's suggestion: let $X = \Sigma M \vee \Sigma^2 M$. Since $X$ is a co-$H$-space, we can add maps in $[X,X]$. Let $g:X\to X$ be the sum of $1_X$ and the map $$ X \to \Sigma M \xrightarrow{\Sigma f} \Sigma^2 M \to X $$ where the first map collapses $\Sigma^2 M$ and the third map is the inclusion of $\Sigma^2 M$. The induced map in homology is $1 + \Sigma f_\ast$. In $H_\ast(X;\mathbf{Z})$, this is $1$. However, in $H_3(X;\mathbf{Z}/2)$, the map $g$ is not the identity, since $\Sigma f_\ast$ is nonzero. If we fix the basis of $H_3(X;\mathbf{Z}/2)$ given by the wedge decomposition of $X$, then $g_\ast$ is represented by the matrix

$$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}. $$

Answer by Sam Isaacson for Counterexamples in algebraic topology?

2011-02-14T07:08:11Z

This is a great question. Here are two of my favorite counterexamples:

Rector proved in 1971 that there are uncountably many complexes $X$ (distinct in the homotopy category) such that $\Omega X \simeq S^3$.
It's possible to construct "ghost" maps $f:X\to Y$ that are zero on $\pi_\ast$, but nonetheless essential (e.g., maps of Eilenberg-Mac Lane spaces representing cohomology operations). Even wilder, there are "phantom" maps $f:X\to Y$ so that if $K$ is any finite complex and $i:K\to X$ a map, $fi \simeq \ast$, but $f \not\simeq \ast$. Dan Christensen has written extensively about phantom maps in the stable homotopy category; I think the first example of a phantom map is due to Adams and Walker.

Answer by Sam Isaacson for Is there an interesting definition of a category of test categories?

2010-05-21T17:34:06Z

I'm not sure what “interesting” means in this context. It's probably too much to demand that morphisms between $\widehat{C_1} = \mathbf{Set}^{{C_1}^{\mathrm{Op}}}$ and $\widehat{C_2} = \mathbf{Set}^{{C_2}^{\mathrm{Op}}}$ arise only from functors $C_2 \to C_1$. This would eliminate functors such as the simplicial realization of a cubical set. Better, we should take the morphisms among left adjoints $\widehat{C_1} \to \widehat{C_2}$, i.e., diagrams $C_1 \times {C_2}^\mathrm{Op} \to \mathbf{Set}$. Not all of these give Quillen adjunctions, but those corresponding to suitably cofibrant resolutions of the terminal object in ${\widehat{C_2}}^{C_1}$ probably do.

In the special case that a left adjoint $\widehat{C_1}\to\widehat{C_2}$ is given by restriction $f^\ast$ along an aspherical functor $f:C_2 \to C_1$, the corresponding functor $C_2 \downarrow f^\ast X \to C_1 \downarrow X$ is a weak equivalence for all $X\in\widehat{C_1}$. This is one implication of Proposition 1.2.9 in Maltsiniotis' Asterisque volume. Now, if $f^\ast$ is to be a left Quillen equivalence, you need it to preserve all weak equivalences (since everything in $\widehat{C_1}$ is cofibrant). This forces your hand: since the representables in $\widehat{C_1}$ are weakly contractible you need $f^\ast C_1({-},x)$ to be weakly contractible, i.e., the nerve of $f\downarrow x$ should be weakly equivalent to a point.

Answer by Sam Isaacson for Which math paper maximizes the ratio (importance)/(length)?

2010-05-21T01:39:39Z

Here are two and a half papers in homotopy theory:

Dan Kan introduced Kan complexes and the Kan complex approximation functor $\mathrm{Ex}^\infty$ in the three-page 1956 PNAS paper "Abstract Homotopy III" (here is a JSTOR link). I can't resist pointing out his 1958 Trans. Amer. Math Soc. paper "Adjoint Functors"—clearly too long for this contest at 36 pages—where he defines an adjunction of functors on the first page. Here is a link.
The 1966 Quart. J. Math. Oxford paper $K$-theory and the Hopf invariant by Adams and Atiyah is only 8 pages long. I don't have a link to the paper, but here is a MathSciNet link. Adams and Atiyah use the Adams operations in $K$-theory to solve the Hopf invariant one problem. Adams' original proof (using secondary operations) takes 85 pages—of course that paper was extraordinarily fecund in homotopy theory.

Answer by Sam Isaacson for If Erdős is published as Erdös in a paper, which do I cite?

2010-05-15T07:46:01Z

This is a matter of convention. One guideline is in paragraph 17.20 of the Chicago Manual of Style (15th ed):

Authors' names are normally given as they appear in the title pages of their books. Certain adjustments, however, may be made to assist correct identification (unless they conflict with the style of a particular journal or series. First names may be given in full place of initials. If an author uses his or her given name in one cited book in and initials in another (e.g., "Mary L. Jones" versus "M. L. Jones"), the same form, preferably the fuller one, should be used in all references to that author.

I would err on the side of consistency. Some bibliography styles in LaTeX/bibtex replace subsequent references to the same author with an em dash. If you use many different spellings of the same author's name, this behavior will break.

Answer by Sam Isaacson for Dyer-Lashof based spectral sequence for homotopy classes of maps between infinite loop spaces (spectra).

2010-05-11T05:16:10Z

I'm not sure if this is what you want, but Haynes Miller constructs a spectral sequence computing the homology of a connective spectrum $E$ from the homology of $E_0$ as a Hopf algebra over the Dyer-Lashof algebra in the 1978 Pacific Journal of Mathematics paper "A spectral sequence for the homology of an infinite delooping."

Comment by Sam Isaacson

2012-02-24T18:46:38Z

Interesting! Is there an analogous family for odd primes?

Comment by Sam Isaacson

2011-03-29T05:53:28Z

Sorry; that should be $X = M \vee \Sigma M$

Comment by Sam Isaacson

2011-03-29T05:52:51Z

@Mariano, thanks for pointing out the subtlety I missed. I think you can produce a counterexample stably by looking at $X = M \wedge \Sigma M$ where $M$ is the mod $2$ Moore spectrum. Let $f:X \to X$ be the map which on $\Sigma M$ is the inclusion of the $\Sigma M$ summand and on $M$ is the sum of the inclusion of the $M$ summand and the essential composition $g:M\to S^1 \to \Sigma M$. The map $f$ is the identity on $H\mathbb{Z}$, but since $g$ is nonzero on $H\mathbb{Z}/2$, the map $f$ is not the identity on $H\mathbb{Z}/2$.

Comment by Sam Isaacson

2011-03-29T04:46:31Z

BTW, the simplest example of the failure of the UCT to split naturally that I know of is the map from a mod $n$ Moore space to a sphere that collapses the bottom cell.

Comment by Sam Isaacson

2011-03-29T04:43:10Z

Doesn't the UCT imply that no such counterexample exists? The splitting isn't natural, but the short exact sequence is. More simply, the multiplication by $2$ map on $\mathbb{Z}$ induces a LES in the homology of $X$ and if $f$ is a homology iso, the 5 lemma shows it is an iso with mod $2$ coefficients. Or am I missing something?

Comment by Sam Isaacson

2011-03-09T18:48:21Z

Have you looked at the surveys by Carlsson or Harer and Edelsbrunner? There are a lot of resources at comptop.stanford.edu

Comment by Sam Isaacson

2011-02-23T20:11:33Z

Did you have a specific proof in mind? Also, I believe there is a typo in your statement (consider $n = 1$, $k = 1$). A more general statement is that if $X$ is $n-1$ connected and $n \ge 2$, then the suspension map $\pi_k(X) \to \pi_{k+1}(\Sigma X)$ is an iso if $k\le 2n-2$ and an epimorphism if $k\le 2n-1$; in the case of spheres, $\pi_{n+k}(S^k) \to \pi_{n+k+1}(S^{k+1})$ is an iso if $n+2 \le k$.

Comment by Sam Isaacson

2011-02-23T15:35:30Z

@Harry, combinatorial usually means "locally presentable" and "cofibrantly generated"; did I not make that clear in my comment? Regarding model structures on symmetric spectra, you might add Schwede's excellent "Untitled book project ..." (found on his website) to a list of references. The abundance of model structures on symmetric spectra is a good thing.

Comment by Sam Isaacson

2011-02-23T07:03:01Z

@Harry, the underlying category of symmetric spectra is not precisely "simplicial symmetric sequences" (rather it is modules over the sphere spectrum symmetric sequence). However, the category of symmetric spectra is locally presentable and the model structures constructed in, e.g. HSS, MMSS, Shipley's "A Convenient ..." paper are all cofibrantly generated, hence combinatorial. Although the combinatorial condition is the "right" condition for this sort of thing, for spectra, you don't need all this machinery. You might just want section 2 of Hovey-Palmieri-Strickland.

Comment by Sam Isaacson

2011-02-14T18:28:25Z

Yes; these spaces can be distinguished by higher cohomology operations. You can produce stable counterexamples to #1 quite generally by looking at the cofibers of maps $S^n \to $S^0$ with Adams filtration greater than 1.

Comment by Sam Isaacson

2011-02-14T18:24:54Z

@Henrik, thanks for the correction. @Lennart, you're right; $p$-locally, the only delooping of $S^3$ is $\mathbf{H}\mathrm{P}^\infty$.

Comment by Sam Isaacson

2011-02-14T01:52:27Z

For #1, what about a $\mathbf{Z}/4$ Moore space versus a $\mathbf{Z}/8$ Moore space?

Comment by Sam Isaacson

2010-12-21T20:21:52Z

@Aaron: secondary operations cannot be computed from the unstable $A$-module structure on $H^\ast X$ alone (here $H = H\mathbf{F}_2$. For example, $H^\ast(\Sigma \mathbf{CP}^3)$ and $H^\ast(S^7 \vee \Sigma \mathbf{CP}^2)$ have isomorphic cohomology over $A$, but you can distinguish between these spaces by a secondary operation (this example is in Harper's book on the subject). What I was trying to get at was whether the work of Baues, Jibladze, Nassau, et al (and Adams!), describing the $E_3$ term of the Adams spectral sequence is "algebraic" in either a precise or a moral sense.

Comment by Sam Isaacson

2010-12-18T18:35:18Z

It is not true that the "finest structure we can impose" on $E$-cohomology is that of being a module over $E^\ast E$. For one thing, if $E$ is a ring spectrum, then $E^\ast X$ is a ring; but more subtly, $E^\ast X$ will support "secondary" operations on certain admissible subsets of its cohomology. The simplest example are higher Bockstein operations: if $Sq^1 c = 0$, then $c$ lifts to a class in with $\mathbf{Z}/4$ coefficients; then there is a secondary operation detecting whether $c$ can be lifted to $\mathbf{Z}/8$. However, this formalism might not be algebraic in the sense below.

Comment by Sam Isaacson

2010-09-08T18:42:55Z

Harry, have you looked at Mike Shulman's paper "Homotopy limits and colimits and enriched homotopy theory"? In it, Shulman works out an enriched version of the homotopical machinery of Dwyer, Hirschhorn, Kan, and Smith; some things actually simplify in the enriched setting, since you can work explicitly with the spatial enrichment that is "secretly" there. Anyway, left and right derived functors from $C$ to $Ho(D)$ are right and left Kan extensions (respectively) along $C\to Ho(C)$.