#milnor

So, when you study homotopy theory (and especially when you get into the stable segment of it...) you find yourself saying things like "the homology of homology is the ring of functions on the space of additive polynomials". And things like "this is a Hopf ring with co-product representing polynomial composition". Because that's what Milnor's basis for the dual steenrod algebras actually says ... sort-of. An additive polynomial, by-the-by, looks like a polynomial $P(t) = a_0 + a_1 t + ... a_k t^k $, and has the property $ P(t+s) = P(t) + P(s) $, which means most of the $a_k$ are zero, except for $a_{p^j}$ which must be torsion of order $p$ in whatever ring it inhabits. Yes, we are repeating ourselves somewhat. For $p$ an odd prime, Milnor's basis for $A_p^{\vee}$ is generated by two series, $\xi_j \in H_{2(p^j-1)}(H/p,\mathbb{Z}/p)$ for $j = 1,\dots$ and $\tau_j \in H_{2p^j-1}(H/p,\mathbb{Z}/p)$ for $j=0,\dots$, and says $A_p^{\vee}$ is $$ \Lambda[\tau_j]\otimes \mathbb{F}_p[\xi_j] $$ with homology mod-$p$ bockstein generated by $ \beta\tau_j = \xi_j $ and coproduct $$ \nabla \xi_j = \sum \xi_{l}\otimes \xi_{j-l}^{p^l} $$ $$ \nabla \tau_j = \sum \tau_{l} \otimes \xi_{j-l}^{p^l} $$ but if we invent a variable $t$ (of degree $-2$) then we can write $$ \Xi[t] =\sum \xi_j t^{p^j} $$ (of degree -2) and $$ (\nabla \Xi)[t] = (\Xi\otimes 1)\circ(1\otimes\Xi)[t] $$ and also $$ T[t] = \sum \tau_j t^{p^j} $$ (of degree -1) and $$ (\nabla T)[t] = (T\otimes 1)\circ(1\otimes\Xi)[t] $$ not to mention verifying that $\beta\nabla T = \nabla (\beta T)$. That's the mod-$p$ homology of mod-$p$ homology; things is complicated and tricky, but long story short, in the integral homology of integral homology, you get to keep the just the $\xi_j\in\mathrm{tor}( \ZZ/p H_{2(p^j-1)}(H))$, which somewhere else I called $\Theta_{p^j}$. Weird story: in a middle step, integral homology of mod-$p$ homology (which is the same (but I don't know about ring structures) as the mod-$p$ homology of integral homology) you lose only the generator $\tau_0$. The mod-$2$ case is, of course, special... the generators are like $\xi$ in that they're all polynomial-generators, and they're like $\tau$ in that they're in degrees $2\,2^j-1$. Now some texts (e.g. McCleary) give the coproduct transposed, $\nabla' \xi_j = \sum \xi_l^{j-l} \otimes \xi_l $, which... for me at least... obscures the polynomial composition structure. Maybe that's buried in an exercise (I 〈gasp〉have not done all the exercises...) Mind you, I still don't know what to do with this Thing. But, well, there it is.

MILLLLLL 🥺🥺🥺🥺🥺

Milnor | Logo on an industrial washing machine, laundry

:] :)

One of my favorite photographers, Daniel Milnor, is embarking on a special adventure. He is joining Fredric Roberts Photography Workshops, in association with Save the Children, introducing high school students around the world to photography. The first workshop is currently under way in Bhutan. Do read about it in Dan’s Smogranch blog.