# Second homotopy of a torus complement in the 4-sphere

Let $T$ be the boundary of a solid torus in $S^4$. Are there any theorems or methods which would help one to compute $\pi_2(S^4 -T)$? Or to say if, e.g., it had finite rank and no torsion?
More generally, suppose we have a closed manifold $X$ and we remove a submanifold $Y$ of at least codimension 2. Is there a general method for computing the homotopy groups of $X-Y$? The above case was the original example that got me to think about this but a more general result would also perhaps be helpful.

-
Is your $T$ topologically equal to $S^1\times S^1$? –  Włodzimierz Holsztyński Feb 14 '13 at 4:46
If $X$ is the universal covering space of $S^4 - T$, $H_2 X$ is isomorphic to $\pi_2(S^4 - T)$, this is just the fact that $X$ is simply connected, together with the Hurewicz theorem that says the first non-trivial homotopy and homology groups agree above dimension $2$, and that covering spaces preserve higher homotopy groups. This is a standard technique to compute homotopy groups, for example, Serre made great mileage out of this in his dissertation. –  Ryan Budney Feb 14 '13 at 5:35
See also Sam Lomonaco's Pacific Math Journal paper. Here he is dealing with spheres, but the crossed module structure of $\pi_2$ should be interesting to you as well. –  Scott Carter Feb 15 '13 at 1:40

Note that for links in $\mathbb R^3$ it is standard to use the fundamental group and a Seifert-van Kampen Theorem. So for the case in question this paper uses the fundamental crossed module and a $2$-dimensional Seifert-van Kampen Theorem.
{"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": 1, "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, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8225982189178467, "perplexity": 175.69074960927526}, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644064919.24/warc/CC-MAIN-20150827025424-00276-ip-10-171-96-226.ec2.internal.warc.gz"}