## Schur multipliers for Lie groups?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

In the interest of pursuing the analogies between finite groups and (finite-dimensional) Lie groups, it seems natural to call the Schur multiplier of a finite group analogous to the fundamental group of a Lie group. Just as the Schur multiplier limits what groups can arise as central subgroups of a group $G$ with fixed (isomorphism type of) $Inn(G)$, does the fundamental group do the same thing for Lie groups?

One baby example of this question: If $G$ is a Lie group with a central subgroup $Z$ with $|Z|=4$ and $G/Z \cong SO(3)$, does it follow that $G$ has a subgroup of index $2$?
Also, in the above situation, if $G$ has no subgroup isomorphic to $SO(3)$ and $Z$ is cyclic, does it follow that $G \cong H$, where $H$ is the subgroup of $U(2)$ obtained by adjoining $i$ times the identity matrix to $SU(2)$?

-

2
The Schur multiplier is the second homology, so don't you want to take the second homology of the classifying space $BG$ instead? – Qiaochu Yuan May 24 2011 at 21:48
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 17, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9113983511924744, "perplexity_flag": "head"}