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)


1. the algebraic homology/group homology $H\_{g,\*}(L)$ which is also topological homology of classifying space of discrete underlying group of $L$, $H\_\*(BL^\delta)$;
2. ordinary topological homology of the Lie group as a topological space $H\_\*(L)$;
3. 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.



Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
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.


