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

Question

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.

Answer 1

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.

Answer 2

For future record, since my post question seems already kinda messy, I will document something new in this answer and leave the question itself untouched. The following is trying to answer my question 1, especially when is $BG$ a classifying space of a small category.

In the paper: Segal, Graeme. Classifying spaces and spectral sequences. Publications Mathématiques de l'IHÉS, Volume 34 (1968), pp. 105-112. the author constructed the so called classifying space of a small category as the simplicial geometric realization of the nerve of a prescribed small category. However, a typo in section 3 sorta misleads me into proposing this question. The first sentence in 2nd paragraph should be "The space $\mathbf BG$ is often a classifying space of $G$ in the usual sense..." where $\mathbf BG$ is the classifying space of the singleton category with morphisms parametrized by $G$. Notice this category is a topological category.

If $G$ is an absolute neighborhood retract in $\Delta^n\times G\times\dots\times G$, then $\mathbf BG$ is (a model of) the classifying space $BG$ of $G$. This happens for example when $G$ is a Lie group or discrete group or locally finite CW-complex.

The difference between the topologies of $BG$ and $BG^\delta$ can be seen via the construction of $\mathbf BG,\mathbf BG^\delta$ in the following sense: $BG=\bigsqcup_n\Delta^n\times G^n/\sim, BG^\delta=\bigsqcup_n\Delta^n\times(G^\delta)^n/\sim$, where two $\sim$'s are set theoretically same equivalence relations. Therefore the different topologies of $BG$ and $BG^\delta$ come exactly from $\Delta^n\times G^n$ and $\Delta^n\times(G^\delta)^n$.