Editing Homotopy groups of spheres (section)

===Table===
The following table gives an idea of the complexity of the higher homotopy groups even for spheres of dimension 8 or less. In this table, the entries are either a) the [[trivial group]] 0, the infinite cyclic group {{math|Z}}, b) the finite [[cyclic group]]s of order {{mvar|n}} (written as {{math|Z<sub>''n''</sub>}}), or c) the [[direct product of groups|direct products]] of such groups (written, for example, as {{math|Z<sub>24</sub>×Z<sub>3</sub>}} or {{math|1=Z{{su|lh=1|b=2|p=2}} = Z<sub>2</sub>×Z<sub>2</sub>}}). Extended tables of homotopy groups of spheres are given [[#Table of homotopy groups|at the end of the article]].

{| class="wikitable" style="text-align:center"
|-
!
!style="width:4em"| π<sub>1</sub>
!style="width:4em"| π<sub>2</sub>
!style="width:4em"| π<sub>3</sub>
!style="width:4em"| π<sub>4</sub>
!style="width:4em"| π<sub>5</sub>
!style="width:4em"| π<sub>6</sub>
!style="width:4em"| π<sub>7</sub>
!style="width:4em"| π<sub>8</sub>
!style="width:4em"| π<sub>9</sub>
!style="width:4em"| π<sub>10</sub>
!style="width:4em"| π<sub>11</sub>
!style="width:4em"| π<sub>12</sub>
!style="width:4em"| π<sub>13</sub>
!style="width:4em"| π<sub>14</sub>
!style="width:4em"| π<sub>15</sub>
|-

!style="height: 2.5em"| ''S''<sup>1</sup>
|style="background:white"| Z
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|-
!style="height: 2.5em"| ''S''<sup>2</sup>
|style="background:#FFDDDD; border-top: solid black 2px"| 0
|style="background:#DDDDFF; border-top: solid black 2px; border-right: solid black 2px"| Z
|style="background:#FFFFCC"| Z
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>12</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>3</sub>
|style="background:white"| Z<sub>15</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z<sub>12</sub>×Z<sub>2</sub>
|style="background:white"| Z<sub>84</sub>×Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|-
!style="height: 2.5em"| ''S''<sup>3</sup>
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF; border-top: solid black 2px"| Z
|style="background:#DDFFDD; border-top: solid black 2px; border-right: solid black 2px"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>12</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>3</sub>
|style="background:white"| Z<sub>15</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z<sub>12</sub>×Z<sub>2</sub>
|style="background:white"| Z<sub>84</sub>×Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|-
!style="height: 2.5em"| ''S''<sup>4</sup>
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD; border-top: solid black 2px"| Z<sub>2</sub>
|style="background:#FFDDDD; border-top: solid black 2px; border-right: solid black 2px"| Z<sub>2</sub>
|style="background:#FFFFCC"| Z×Z<sub>12</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z<sub>24</sub>×Z<sub>3</sub>
|style="background:white"| Z<sub>15</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=3}}
|style="background:white"| {{su|p=Z<sub>120</sub>×|b=Z<sub>12</sub>×Z<sub>2</sub>}}
|style="background:white"| Z<sub>84</sub>×Z{{su|lh=1|b=2|p=5}}
|-
!style="height: 2.5em"| ''S''<sup>5</sup>
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD"| Z<sub>2</sub>
|style="background:#FFDDDD; border-top: solid black 2px"| Z<sub>2</sub>
|style="background:#DDDDFF; border-top: solid black 2px; border-right: solid black 2px"| Z<sub>24</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>30</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=3}}
|style="background:white"| Z<sub>72</sub>×Z<sub>2</sub>
|-
!style="height: 2.5em"| ''S''<sup>6</sup>
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD"| Z<sub>2</sub>
|style="background:#FFDDDD"| Z<sub>2</sub>
|style="background:#DDDDFF; border-top: solid black 2px"| Z<sub>24</sub>
|style="background:#DDFFDD; border-top: solid black 2px; border-right: solid black 2px"| 0
|style="background:#FFFFCC"| Z
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>60</sub>
|style="background:white"| Z<sub>24</sub>×Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=3}}
|-
!style="height: 2.5em"| ''S''<sup>7</sup>
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD"| Z<sub>2</sub>
|style="background:#FFDDDD"| Z<sub>2</sub>
|style="background:#DDDDFF"| Z<sub>24</sub>
|style="background:#DDFFDD; border-top: solid black 2px"| 0
|style="background:#FFDDDD; border-top: solid black 2px; border-right: solid black 2px"| 0
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>120</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=3}}
|-
!style="height: 2.5em"| ''S''<sup>8</sup>
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD"| Z<sub>2</sub>
|style="background:#FFDDDD"| Z<sub>2</sub>
|style="background:#DDDDFF"| Z<sub>24</sub>
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD; border-top: solid black 2px"| 0
|style="background:#DDDDFF; border-top: solid black 2px; border-right: solid black 2px"| Z<sub>2</sub>
|style="background:#FFFFCC; border-bottom: solid black 2px"| Z×Z<sub>120</sub>
|}

The first row of this table is straightforward. The homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>1</sup>)}} of the 1-sphere are trivial for {{math|''i'' > 1}}, because the universal [[covering space]], <math>\mathbb{R}</math>, which has the same higher homotopy groups, is contractible.{{sfn|Hatcher|2002|p=342}}

Beyond the first row, the higher homotopy groups ({{math|''i'' > ''n''}}) appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle.
* The groups below the jagged black line are constant along the diagonals (as indicated by the red, green and blue coloring).
* Most of the groups are finite.  The only infinite groups are either on the main diagonal or immediately above the jagged line (highlighted in yellow).
* The second and third rows of the table are the same starting in the third column (i.e., {{math|π<sub>''i''</sub>(''S<sup>2</sup>'') {{=}} π<sub>''i''</sub>(''S<sup>3</sup>'')}} for {{math| ''i'' ≥ 3}}).  This isomorphism is induced by the Hopf fibration {{math| ''S''<sup>3</sup> → ''S''<sup>2</sup>}}.
* For {{math|1=''n'' = 2, 3, 4, 5}} and {{math|''i'' ≥ ''n''}} the homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>)}} do not vanish. However, {{math|1=π<sub>''n''+4</sub>(''S''<sup>''n''</sup>) = 0}} for {{math|''n'' ≥ 6}}.
These patterns follow from many different theoretical results.{{cn|date=February 2022}}