## Third bordism group of BG, where G is an arbitrary compact Lie group.

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

Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ consists completely of torsion when $G$ is compact and I would like (if possible) a similar type of statement for $\Omega_3(BG)$.

-

## 1 Answer

If you think about oriented bordism, the answer is that $\Omega_3 (BG) \cong H_3 (BG)$. This is true for any space $X$ instead of $BG$, because of the Atiyah-Hirzebruch spectral sequence and because $\Omega_i (pt)=0$ for $i=1,2,3$.

-

Thank you! Can you say anything as nice for the unoriented case? – Kevin Wray Apr 29 2011 at 22:30
Is it obvious that Atiyah-Hirzebruch applies to the associated homology of a spectrum in the same way as the associated cohomology? – Dylan Wilson Apr 30 2011 at 2:54
@Dylan: I think the exact couple set up makes it reasonably clear. Have you looked at Adams? – Sean Tilson Apr 30 2011 at 3:30
The unoriented case is even simpler as unoriented bordism is just homology with coefficients in the (unoriented) bordism ring. – Torsten Ekedahl Apr 30 2011 at 6:57
Wait, I thought I'd read somewhere that $\Omega_3(BG)$ was equivalent to $H_3(BG)$ up to torsion.  – Kevin Wray Apr 30 2011 at 19:09
{"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": 13, "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": 0.9443656206130981, "perplexity_flag": "middle"}