1
$\begingroup$

In Adams' lectures on generalised cohomology Page 51, it states $E^*(E)$ is a topology ring. I do not know the topology on it. the reference that Adams offered there is by Novikov in Russian. Can anyone help to explain it?

Thanks!

$\endgroup$
2
$\begingroup$

Suppose we have spectra $X$ and $Y$. For any finite spectrum $W$, and any maps $X\xleftarrow{i}W\xrightarrow{f}Y$, we define $$ N(i,f) = \{g\colon X\to Y \;|\; gi=f\} \subseteq [X,Y]. $$ Sets of this type form a basis for a topology on the set $[X,Y]$. If $X$ is a CW spectrum (as it always is in Adams's book) then you can restrict attention to the case where $W$ is a finite subspectrum of $X$ and $i$ is the inclusion; this will give the same topology. Note that when $X$ itself is finite the topology is discrete, so the definition is only really interesting in the infinite case.

$\endgroup$
  • $\begingroup$ Thank you very much. I hope you do not mind me asking a few quesitons about it. 1 Do I need to module the homotopy equivalence in [X,Y]? If so, is it obvious to see the definition of N(i,f) is compatible with the quotient. 2 Can I think this one as an analogue to the open compact topology for mapping space? 3 Why this topology is important in the infinite case?(I need to consider it when using Adams Novikov Spectral Sequence in cohomology)? Thanks again! $\endgroup$ – user48537 Apr 28 '14 at 23:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.