Thursday, January 31, 2008

Naive Maps of Spectra

Being a math grad student, I figure I should occasionally post some math up here. And since I learned something cool in class today, I thought I'd talk about that.

So, suppose that $E_0$, $E_1$, $F_0$, and $F_1$ are pointed topological spaces (maybe I should say CW complexes throughout?) and we have maps $\sigma_E:\Sigma E_0\rightarrow E_1$ and $\sigma_F:\Sigma F_0\rightarrow F_1$. Let $\sigma_E^{ad}$ denote the adjoint map $E_0\rightarrow \Omega E_1$, and similarly for $\sigma_F^{ad}$. For pointed topological spaces $X$ and $Y$, let $Top_*(X,Y)$ denote the (pointed) space of based maps from $X$ to $Y$.

Using these structure maps (the $\sigma$s and/or their adjoints), we get maps $Top_*(E_0,F_0)\rightarrow Top_*(E_0,\Omega F_1)$ and $Top_*(E_1,F_1)\rightarrow Top_*(\Sigma E_0,F_1)$. Recall that these two targets are naturally isomorphic (=homeomorphic here). So (allowing me some sloppiness (if I were using full LaTeX, I'd put a nice xymatrix here)), we can fit these spaces into a diagram $A\rightarrow B\leftarrow C$. Then, playing with adjoints and things, it isn't too hard to see that the limit (=pullback) of this diagram is the collection of pairs of maps $f_0:E_0\rightarrow F_0$ and $f_1:E_1\rightarrow F_1$ such that $\sigma_F\circ \Sigma f_0=f_1\circ \sigma_E$.

This is the first step in our naive maps of spectra. Recall that a spectrum $E_*$ is a sequence $\{E_i\}_{i=0}^{\infty}$ of spaces together with structure maps $\sigma_i:\Sigma E_i\rightarrow E_{i+1}$. A naive map between two such spectra ($E_*$ and $F_*$) is then a sequence $\{f_i:E_i\rightarrow F_i\}_{i=0}^{\infty}$ of maps of spaces that commute with the structure maps (in the obvious sense (its fun to just say that and let you sort it out :) )). Generalizing from our above work, we can describe this as the limit over a diagram $A_0\rightarrow A_{01}\leftarrow A_1\rightarrow A_{12}\leftarrow A_2\rightarrow\cdots$, where $A_i$ is $Top_*(E_i,F_i)$ and $A_{ij}$ is $Top_*(\Sigma E_i,F_{i+1})\cong Top_*(E_i,\Omega F_{i+1})$.

The reason one calls these maps 'naive' is that there is another notion of maps between spectra that works better for homotopy theory. Adams says the motto is 'cells now, maps later', which we might interpret here has saying that it doesn't really matter if we start defining maps at the 0 part of the spectrum. As long as we eventually have levelwise maps, that should be good enough.

I've always found the definitions of spectra and maps between them to be pretty thorny. While I'm not saying the above helps, necessarily, it was a fun thing to see, playing with limits. I guess the main thing to take away from this is that anytime you describe a space as a subspace of a product space (we've got maps of spectra being a subspace of the product of all levelwise maps), you might try to think about it as a limit construction.