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Homotopy of torus - Revision history
2026-09-08T15:34:34Z
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https://topospaces.subwiki.org/w/index.php?title=Homotopy_of_torus&diff=3570&oldid=prev
Vipul at 21:01, 2 April 2011
2011-04-02T21:01:04Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:01, 2 April 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Relation with universal covering space==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Relation with universal covering space==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The universal covering space of the torus <math>T^n</math> is [[Euclidean space]] <math>\R^n</math>, and in fact <math>T^n \cong \R^n/\mathbb{Z}^n</math> where the latter is the lattice of points with integer coordinates. The universal cover is a [[contractible space]], and this is equivalent to the observation that <math>T^n</math> is <del style="font-weight: bold; text-decoration: none;">an </del>aspherical space.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The universal covering space of the torus <math>T^n</math> is [[Euclidean space]] <math>\R^n</math>, and in fact <math>T^n \cong \R^n/\mathbb{Z}^n</math> where the latter is the lattice of points with integer coordinates. The universal cover is a [[contractible space]], and this is equivalent to the observation that <math>T^n</math> is <ins style="font-weight: bold; text-decoration: none;">a path-connected </ins>aspherical space.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In particular, <math>T^n</math> can be viewed as a [[classifying space]] for the infinite discrete group <math>\mathbb{Z}^n</math>.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Facts used in computation==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Facts used in computation==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># [[Homotopy of spheres]] (in particular, homotopy groups of the [[circle]])</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># [[Homotopy of spheres]] (in particular, homotopy groups of the [[circle]])</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># [[Homotopy group of product is product of homotopy groups]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># [[Homotopy group of product is product of homotopy groups]]</div></td></tr>
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Vipul
https://topospaces.subwiki.org/w/index.php?title=Homotopy_of_torus&diff=3569&oldid=prev
Vipul at 21:00, 2 April 2011
2011-04-02T21:00:04Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:00, 2 April 2011</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l12">Line 12:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Case <math>k = 1</math>: The [[fundamental group]] <math>\pi_1(T^n)</math> is the group <math>\mathbb{Z}^n</math>, i.e., the product of <math>n</math> copies of the infinite cyclic group. In other words, it is the free abelian group of rank <math>n</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Case <math>k = 1</math>: The [[fundamental group]] <math>\pi_1(T^n)</math> is the group <math>\mathbb{Z}^n</math>, i.e., the product of <math>n</math> copies of the infinite cyclic group. In other words, it is the free abelian group of rank <math>n</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Case <math>k \ge 2</math>: Any higher homotopy group <math>\pi_k(T^n)</math> is the [[trivial group]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Case <math>k \ge 2</math>: Any higher homotopy group <math>\pi_k(T^n)</math> is the [[trivial group]].</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In particular, this means that any torus is an [[aspherical space]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Relation with universal covering space==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The universal covering space of the torus <math>T^n</math> is [[Euclidean space]] <math>\R^n</math>, and in fact <math>T^n \cong \R^n/\mathbb{Z}^n</math> where the latter is the lattice of points with integer coordinates. The universal cover is a [[contractible space]], and this is equivalent to the observation that <math>T^n</math> is an aspherical space.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Facts used in computation==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Facts used in computation==</div></td></tr>
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Vipul
https://topospaces.subwiki.org/w/index.php?title=Homotopy_of_torus&diff=3568&oldid=prev
Vipul: Created page with "{{homotopy invariant computation| invariant = homotopy group| space = torus}} This article gives the key facts about the computation of homotopy groups for the <math>n</math..."
2011-04-02T20:58:22Z
<p>Created page with "{{homotopy invariant computation| invariant = homotopy group| space = torus}} This article gives the key facts about the computation of <a href="/wiki/Homotopy_group" title="Homotopy group">homotopy groups</a> for the <math>n</math..."</p>
<p><b>New page</b></p><div>{{homotopy invariant computation|<br />
invariant = homotopy group|<br />
space = torus}}<br />
<br />
This article gives the key facts about the computation of [[homotopy group]]s for the <math>n</math>-torus <math>T^n</math>, defined as the product of <math>n</math> copies of the [[circle]].<br />
<br />
==Statement==<br />
<br />
<math>\pi_k(T^n)</math> is given as follows:<br />
<br />
* Case <math>k = 0</math>: The [[set of path components]] <math>\pi_0(T^n)</math> is the one-point set, and we can think of it as the [[trivial group]].<br />
* Case <math>k = 1</math>: The [[fundamental group]] <math>\pi_1(T^n)</math> is the group <math>\mathbb{Z}^n</math>, i.e., the product of <math>n</math> copies of the infinite cyclic group. In other words, it is the free abelian group of rank <math>n</math>.<br />
* Case <math>k \ge 2</math>: Any higher homotopy group <math>\pi_k(T^n)</math> is the [[trivial group]].<br />
<br />
==Facts used in computation==<br />
<br />
# [[Homotopy of spheres]] (in particular, homotopy groups of the [[circle]])<br />
# [[Homotopy group of product is product of homotopy groups]]</div>
Vipul