I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological groups with the same underlying groups.
My question 1 is: Are those classifying spaces associated to different topological groups (with the same underlying groups) the same? (For this question my guess is no)
For example, $G=\mathbb{Z}/2$ is a group. $X$ is the topological group $G$ with discrete topology, $Y$ is $G$ equipped with trivial topology (only open sets are empty set and the whole set). Then $X$ is disjoint union of singletons, $Y$ is contractible. What are classifying spaces $BX,BY$? Are they the same? I know $BX$ may be chosen as $\mathbb{RP}^\infty$, what about $BY$?
It is well-known that if $G$ is discrete, then $BG$ is the classifying space of a small category or EM space $K(G,1)$. What if $G$ is not discrete, and if the answer to my question 1 is no, then is $BG$ still the classifying space of some small category?
My question 2 (if the answer to question 1 is no) is: When are these different classifying spaces comparable or identical? To be more specific, if $G,H$ are different topological groups with the same underlying groups and a continuous group isomorphism $G\to H$ (e.g. $G$ is discrete), do we have a principal bundle morphism? That is maps $EG\to EH, BG\to BH$ that commute with bundle maps. Is there any criterion for these maps to be homotopy equivalences?
A corollary of question 2 is to understand what can be said about the relationships between group homology of a Lie group and homology of its classifying space.
My question 3 is: Given a (connected) Lie group $L$, we have three interesting homologies (with coefficients to be determined)
- the algebraic homology/group homology $H_{g,*}(L)$ which is also topological homology of classifying space of discrete underlying group of $L$, $H_*(BL^\delta)$;
- ordinary topological homology of the Lie group as a topological space $H_*(L)$;
- topological homology of the classifying space of $L$, $H_*(BL)$.
I'm wondering what is the relation between them.
For 2 and 3 we have Eilenberg-Moore spectral sequence $E^2_{p,q}=\text{Tor}_{p,q}^{H_*(L;k)}(k,k)\Rightarrow H_{p+q}(BL;k)$ where $k$ is a field and also Serre spectral sequence. If the answer to question 2 is comparable then it would relate 3 to 1.
Background: Homology 1 in question 3 could provide important invariants for scissors congruence theory in non-Euclidean spaces; Homology 2 is important in its own interest; Homology 3 is nothing but characteristic classes.
Best Answer
You may want to look at the classical paper of Jack Milnor, "On the homology of Lie groups made discrete." The Friedlander-Milnor conjecture states that the map $BG^\delta \to BG$ (where $G$ is a Lie group and $G^\delta$ is $G$ made discrete) is a mod $p$ cohomology equivalence. Also chase papers referring to this one, but I think the conjecture is still open. With integer coefficients, this is very false, as the paper explains, by the work of Sah and Wagoner.