Xem mã nguồn của Nhóm đồng luân của các hình cầu

[[Tập tin:Sphere_wrapped_round_itself.png|nhỏ|Quấn một hình cầu hai chiều quanh một hình cầu khác.]]
Trong [[toán học]], và cụ thể hơn là trong [[tô pô đại số]], các '''nhóm đồng luân của hình cầu''' là các bất biến mô tả, một cách đại số, những cách mà các hình cầu <math>n</math> chiều và <math>k</math> chiều có thể quấn quanh nhau. Khái niệm này, vốn ban đầu được xác định cho các mặt cầu 1 chiều (vòng tròn) và 2 chiều (hình cầu), được khái quát cho các mặt cầu <math>n</math> chiều.

== Định nghĩa  ==
Nhóm đồng luân bậc <math>j</math> của [[Mặt cầu|hình cầu]] <math>n</math> chiều <math>\mathbb S^n </math>, là tập hợp, ký hiệu <math>\pi_j(\mathbb S^n) =[\mathbb{S}^j\to\mathbb S^n]</math>, các lớp [[đồng luân]] của các hàm liên tục giữa hai hình cầu sao cho một điểm cố định của [[Mặt cầu|hình cầu]] <math>\mathbb{S}^j</math> được gửi tới một điểm cố định của hình cầu <math>\mathbb S^n</math> (gọi là hai điểm cơ sở).

Tập hợp <math>\pi_j(\mathbb S^n)</math> có thể được trang bị một cấu trúc [[Nhóm giao hoán|nhóm abel]].

Nếu <math>j<n</math>, nhóm này là nhóm tầm thường: <math>\pi_j(\mathbb S^n)=\{0\}</math>.

Nếu <math>j=n</math>, ta có <math>\pi_n(\mathbb S^n)=\Z</math> (có thể chứng minh bằng [[định lý Hurewicz]]).

== 1 chiều: các nhóm đồng luân của đường tròn ==
Ta có:

* <math>\pi_1(\mathbb S^1)=\Z</math>; 
* <math>\pi_q(\mathbb S^1)=0\quad</math>với<math>\quad q\ge2</math>.

== 2 chiều và 3 chiều ==
Các hình cầu có ít nhất hai chiều là [[Không gian đơn liên|đơn liên]], nói riêng:

: <math>\pi_1(\mathbb S^2)=\pi_1(\mathbb S^3)=0</math>

Với mọi <math>n</math> lớn hơn hoặc bằng 3, ta có: <math>\pi_2(\mathbb S^n)=0</math>, nói riêng:

: <math>\pi_2(\mathbb S^3)=0</math>

Với mọi <math>n</math>, ta có: <math>\pi_n(\mathbb S^n)=\Z</math>, nói riêng:

: <math>\pi_2(\mathbb S^2)=\Z</math>, 
: <math>\pi_3(\mathbb S^3)=\Z</math>.

[[Tập tin:Hopfkeyrings.jpg|nhỏ|Biểu diễn ba chiều của một phần thành thớ Hopf]]

Thành thớ Hopf<center><math>F=S^1\hookrightarrow S^3\rightarrow S^2=B \,\!</math></center> cho ta một dãy khớp đồng luân, <center><math> \pi_i(S^1)\to \pi_i(S^3) \to \pi_i(S^2) \to \pi_{i-1}(S^1)\to \pi_{i-1}(S^3)\, \cdots </math></center> Từ <math>\pi_1(\mathbb S^1)=\Z</math> và với <math>i\ge 2</math>, <math>\pi_i(\mathbb S^1)=0</math> ta có một đẳng cấu

: <math>\pi_i(\mathbb S^3)\simeq\pi_i(\mathbb S^2)</math> với <math>i\ge3</math>,

nói riêng

: <math>\pi_3(\mathbb S^2)=\pi_3(\mathbb S^3)=\Z</math>

Với các nhóm đồng luân bậc cao hơn, nhiều kỹ thuật khác cho ta các kết quả sau

* <math>\pi_4(\mathbb S^2)=\pi_4(\mathbb S^3)=\Z/(2)</math>
* <math>\pi_5(\mathbb S^2)=\pi_5(\mathbb S^3)=\Z/(2)</math>
* <math>\pi_6(\mathbb S^2)=\pi_6(\mathbb S^3)=\Z/(12) </math>

{| class="wikitable" style="text-align: center; margin: auto;"
|+ '''Các nhóm đồng luân của <math>\mathbb S^3</math> và <math>\mathbb S^2</math>''' 
| <math>k</math> 
| 3 
| 4 
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| số 8 
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|-
| <math>\pi_k(\mathbb S^3)=\pi_k(\mathbb S^2)</math> 
| '''Z''' 
| colspan="2" | '''Z'''<sub>2</sub> 
| '''Z'''<sub>12</sub> 
| colspan="2" | '''Z'''<sub>2</sub> 
| '''Z'''<sub>3</sub> 
| '''Z'''<sub>15</sub> 
| '''Z'''<sub>2</sub> 
| '''Z'''<sub>2</sub><sup>2</sup> 
| '''Z'''<sub>12</sub>&#xD7;'''Z'''<sub>2</sub> 
| '''Z'''<sub>84</sub>&#xD7;'''Z'''<sub>2</sub><sup>2</sup> 
| '''Z'''<sub>2</sub><sup>2</sup> 
| '''Z'''<sub>6</sub> 
| colspan="2" | '''Z'''<sub>30</sub> 
| '''Z'''<sub>2</sub>&#xD7;'''Z'''<sub>6</sub> 
| colspan="2" | '''Z'''<sub>2</sub><sup>2</sup>&#xD7;'''Z'''<sub>12</sub> 
| '''Z'''<sub>2</sub><sup>2</sup>&#xD7;'''Z'''<sub>132</sub> 
|}

== Chiều cao hơn ==

=== Bảng ===
Tính toán các nhóm đồng luân của các hình cầu nói chung là phức tạp. Bảng sau tóm gọn lại kết quả thu được. 
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!
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! style="width:3em" |&#x3C0;<sub>8</sub>
! style="width:3em" |&#x3C0;<sub>9</sub>
! style="width:3em" |&#x3C0;<sub>10</sub>
! style="width:3em" |&#x3C0;<sub>11</sub>
! style="width:3em" |&#x3C0;<sub>12</sub>
! style="width:3em" |&#x3C0;<sub>13</sub>
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! style="width:3em" |&#x3C0;<sub>15</sub>
! style="width:3em" |&#x3C0;<sub>16</sub>
|-
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| style="background:#FFFFCC" |'''Z'''
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| rowspan="2" style="background:white" |'''Z'''<sub>12</sub>
| rowspan="2" style="background:white" |'''Z'''<sub>2</sub>
| rowspan="2" style="background:white" |'''Z'''<sub>2</sub>
| rowspan="2" style="background:white" |'''Z'''<sub>3</sub>
| rowspan="2" style="background:white" |'''Z'''<sub>15</sub>
| rowspan="2" style="background:white" |'''Z'''<sub>2</sub>
| rowspan="2" style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
| rowspan="2" style="background:white" |'''Z'''<sub>12</sub>&#xD7;'''Z'''<sub>2</sub>
| rowspan="2" style="background:white" |'''Z'''<sub>84</sub>&#xD7;'''Z'''<sub>2</sub><sup>2</sup>
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| rowspan="2" style="background:white" |'''Z'''<sub>6</sub>
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| style="background:#FFCCCC" |'''Z'''&#xD7;'''Z'''<sub>12</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
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| style="background:white" |'''Z'''<sub>24</sub>&#xD7;'''Z'''<sub>3</sub>
| style="background:white" |'''Z'''<sub>15</sub>
| style="background:white" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>120</sub>&#xD7;'''Z'''<sub>12</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>84</sub>&#xD7;'''Z'''<sub>2</sub><sup>5</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>6</sup>
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| style="background:white" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>30</sub>
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| style="background:white" valign="top" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>72</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" valign="top" |'''Z'''<sub>504</sub>&#xD7;'''Z'''<sub>2</sub><sup>2</sup>
|-
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| style="background:white" valign="top" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>72</sub>&#xD7;'''Z'''<sub>2</sub>
|-
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| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>120</sub>
| style="background:white" valign="top" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" valign="top" |'''Z'''<sub>2</sub><sup>4</sup>
|-
! style="height: 2.5em" |''S''<sup>8</sup>
| style="background:white" |0
| style="background:white" |0
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| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF; border-top: solid black 2px" |0
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| style="background:#FFCCCC" |'''Z'''&#xD7;'''Z'''<sub>120</sub>
| style="background:white" valign="top" |'''Z'''<sub>2</sub><sup>4</sup>
|-
! style="height: 2.5em" |''S''<sup>9</sup>
| style="background:white" |0
| style="background:white" |0
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| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
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| style="background:#CCFFFF; border-top: solid black 2px" |'''Z'''<sub>2</sub>
| style="background:#ddf; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>240</sub>
|}

=== Ổn định khi số chiều lớn ===
Bảng đồng luân của <math>\pi_{n+k}(\mathbb S^n)</math> dễ nhìn hơn: 
{| class="wikitable centre" style="text-align:center"
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! style="width:3em" |&#x3C0;<sub>''n''+1</sub>
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! style="width:3em" |&#x3C0;<sub>''n''+3</sub>
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! style="width:3em" |&#x3C0;<sub>''n''+7</sub>
! style="width:3em" |&#x3C0;<sub>''n''+8</sub>
! style="width:3em" |&#x3C0;<sub>''n''+9</sub>
! style="width:3em" |&#x3C0;<sub>''n''+10</sub>
! style="width:3em" |&#x3C0;<sub>''n''+11</sub>
! style="width:3em" |&#x3C0;<sub>''n''+12</sub>
! style="width:3em" |&#x3C0;<sub>''n''+13</sub>
! style="width:3em" |&#x3C0;<sub>''n''+14</sub>
! style="width:3em" |&#x3C0;<sub>''n''+15</sub>
|-
! style="height: 2.5em" |''S''<sup>1</sup>
| style="background:#FFFFCC" |'''Z'''
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| style="background:#FFFFCC; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''
| style="background:#FFCCCC" |'''Z'''
| style="background:#FFCCFF;" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf;" |'''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:#ddf;" |'''Z'''<sub>15</sub>
| style="background:white" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>12</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:#ddf;" |'''Z'''<sub>84</sub>&#xD7;'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>6</sub>
| style="background:white" |'''Z'''<sub>30</sub>
|-
! style="height: 2.5em" |''S''<sup>3</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf;" |'''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:#ddf;" |'''Z'''<sub>15</sub>
| style="background:white" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>12</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:#ddf;" |'''Z'''<sub>84</sub>&#xD7;'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>6</sub>
| style="background:white" |'''Z'''<sub>30</sub>
| style="background:white" |'''Z'''<sub>30</sub>
|-
! style="height: 2.5em" |''S''<sup>4</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>2</sub>
| style="background:#FFCCCC" |'''Z'''&#xD7;'''Z'''<sub>12</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>24</sub>&#xD7;'''Z'''<sub>3</sub>
| style="background:white" |'''Z'''<sub>15</sub>
| style="background:white" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>120</sub>&#xD7;'''Z'''<sub>12</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>84</sub>&#xD7;'''Z'''<sub>2</sub><sup>5</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>6</sup>
| style="background:white" |'''Z'''<sub>24</sub>&#xD7;'''Z'''<sub>6</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>2520</sub>&#xD7;'''Z'''<sub>6</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>30</sub>
|-
! style="height: 2.5em" |''S''<sup>5</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf; 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:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>30</sub>
| style="background:white" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>72</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>504</sub>&#xD7;'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>6</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>6</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>30</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>6</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC; border-top: solid black 2px; border-right: solid black 2px" |0
| style="background:#FFCCCC" |'''Z'''
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>60</sub>
| style="background:white" |'''Z'''<sub>24</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>72</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>504</sub>&#xD7;'''Z'''<sub>4</sub>
| style="background:white" |'''Z'''<sub>240</sub>
| style="background:white" |'''Z'''<sub>6</sub>
| style="background:white" |'''Z'''<sub>12</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>60</sub>&#xD7;'''Z'''<sub>6</sub>
|-
! style="height: 2.5em" |''S''<sup>7</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF; border-top: solid black 2px; border-right: solid black 2px" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>120</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>4</sup>
| style="background:white" |'''Z'''<sub>24</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>504</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |0
| style="background:white" |'''Z'''<sub>6</sub>
| style="background:white" |'''Z'''<sub>24</sub>&#xD7;'''Z'''<sub>4</sub>
| style="background:white" |'''Z'''<sub>120</sub>&#xD7;'''Z'''<sub>2</sub><sup>3</sup>
|-
! style="height: 2.5em" |''S''<sup>8</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>2</sub>
| style="background:#FFCCCC" |'''Z'''&#xD7;'''Z'''<sub>120</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>4</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>5</sup>
| style="background:white" |'''Z'''<sub>24</sub><sup>2</sup>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>504</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |0
| style="background:white" |'''Z'''<sub>6</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>240</sub>&#xD7;'''Z'''<sub>24</sub>&#xD7;'''Z'''<sub>4</sub>
| style="background:white" |'''Z'''<sub>120</sub>&#xD7;'''Z'''<sub>2</sub><sup>5</sup>
|-
! style="height: 2.5em" |''S''<sup>9</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>240</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>2</sub><sup>4</sup>
| style="background:white" |'''Z'''<sub>24</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>504</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |0
| style="background:white" |'''Z'''<sub>6</sub>
| style="background:white" |'''Z'''<sub>16</sub>&#xD7;'''Z'''<sub>4</sub>
| style="background:white" |'''Z'''<sub>240</sub>&#xD7;'''Z'''<sub>2</sub><sup>3</sup>
|-
! style="height: 2.5em" |''S''<sup>10</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCCC" |'''Z'''&#xD7;'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>12</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub>
| style="background:white" |'''Z'''<sub>12</sub>
| style="background:white" |'''Z'''<sub>6</sub>
| style="background:white" |'''Z'''<sub>16</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>240</sub>&#xD7;'''Z'''<sub>2</sub><sup>2</sup>
|-
! style="height: 2.5em" |''S''<sup>11</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:white" |'''Z'''<sub>6</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub>
| style="background:white" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:white" |'''Z'''<sub>6</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>16</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>240</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>12</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>6</sub>
| style="background:#FFCCCC" |'''Z'''&#xD7;'''Z'''<sub>504</sub>
| style="background:white" |'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>6</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>48</sub>&#xD7;'''Z'''<sub>4</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:white" |'''Z'''<sub>240</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>13</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF" |'''Z'''<sub>6</sub>
| style="background:#ddf; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>504</sub>
| style="background:#FFFFCC" |0
| style="background:white" |'''Z'''<sub>6</sub>
| style="background:white" |'''Z'''<sub>16</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>480</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>14</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF" |'''Z'''<sub>6</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub>
| style="background:#FFFFCC; border-top: solid black 2px; border-right: solid black 2px" |0
| style="background:#FFCCCC" |'''Z'''&#xD7;'''Z'''<sub>3</sub>
| style="background:white" |'''Z'''<sub>8</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>480</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>15</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF" |'''Z'''<sub>6</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>3</sub>
| style="background:white" |'''Z'''<sub>4</sub>&#xD7;'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>480</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>16</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF" |'''Z'''<sub>6</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |'''Z'''<sub>3</sub>
| style="background:#CCFFFF; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCCC" |'''Z'''&#xD7;'''Z'''<sub>480</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>17</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF" |'''Z'''<sub>6</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |'''Z'''<sub>3</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#ddf; border-top: solid black 2px; border-right: solid black 2px" |'''Z'''<sub>480</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>18</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF" |'''Z'''<sub>6</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |'''Z'''<sub>3</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#ddf" |'''Z'''<sub>480</sub>&#xD7;'''Z'''<sub>2</sub>
|-
! style="height: 2.5em" |''S''<sup>19</sup>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#FFCCFF" |'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF" |'''Z'''<sub>6</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |'''Z'''<sub>3</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#ddf" |'''Z'''<sub>480</sub>&#xD7;'''Z'''<sub>2</sub>
|}
Với số chiều đủ lớn, ta có

* <math>\pi_n(\mathbb S^n)=\Z,\quad n\ge1</math> (cột đầu tiên màu vàng của bảng trên) 
* <math>\pi_{n+1}(\mathbb S^n)=\Z/(2),\quad n\ge3</math> (cột thứ hai - màu tím - của bảng trên) 
* <math>\pi_{n+2}(\mathbb S^n)=\Z/(2),\quad n\ge2</math> (cột thứ ba - màu lam - của bảng trên)

Hóa ra là <math>\Gamma_k=\pi_{n+k}(\mathbb S^n)</math> không phụ thuộc vào <math>n</math> với <math>n</math> đủ lớn. Hiện tượng này được gọi là sự ổn định. Nó xuất phát từ định lý suspension Freudenthal sau đây:

* ''Các đồng cấu suspension <math>S:\pi_{n+k}(\mathbb S^n)\to\pi_{n+k+1}(\mathbb S^{n+1})</math> là một đẳng cấu với <math>n\ge k+2</math>'' 
* ''và là một toàn cấu (theo nghĩa một đồng cấu toàn ánh) với <math>n=k+1</math>.''

=== Danh sách các nhóm đồng luân ổn định ===
Các nhóm ổn định đầu tiên <math>\Gamma_k=\pi_{2k+2}(\mathbb S^{k+2})=\pi_{n+k}(\mathbb S^n),\quad n\ge k+2</math> là như sau:

* <math>\Gamma_{-j}=\pi_{n-j}(\mathbb S^n)=0,</math>
* <math>\Gamma_0=\pi_n(\mathbb S^n)=\Z,\quad n\ge1</math>
* <math>\Gamma_1=\pi_{n+1}(\mathbb S^n)=\Z/(2),\quad n\ge3</math>
* <math>\Gamma_2=\pi_{n+2}(\mathbb S^n)=\Z/(2),\quad n\ge2</math>
* <math>\Gamma_3=\pi_{n+3}(\mathbb S^n)=\Z/(24),\quad n\ge5</math>
* <math>\Gamma_4=\pi_{n+4}(\mathbb S^n)=0,\quad n\ge6</math>
* <math>\Gamma_5=\pi_{n+5}(\mathbb S^n)=0,\quad n\ge7</math>
* <math>\Gamma_6=\pi_{n+6}(\mathbb S^n)=\Z/(2),\quad n\ge5</math>
* <math>\Gamma_7=\pi_{n+7}(\mathbb S^n)=\Z/(240),\quad n\ge9</math>
* <math>\Gamma_8=\pi_{n+8}(\mathbb S^n)=\Z/(2)\oplus\Z/(2),\quad n\ge10</math>

Các nhóm đồng luân ổn định là hữu hạn ngoại trừ <math>k=0</math>. 
{| class="wikitable" style="text-align: center; margin: auto;"
|+ '''Nhóm đồng luân ổn định'''    <math>\ \Gamma_k=\pi_{2k+2}(\mathbb S^{k+2})</math> với <math>k</math> nhỏ hơn 23 
| <math>k</math> 
| 0 
| 1 
| 2 
| 3 
| 4 
| 5 
| 6 
| 7 
| số 8 
| 9 
| 10 
| 11 
| 12 
| 13 
| 14 
| 15 
| 16 
| 17 
| 18 
| 19 
| 20 
| 21 
| 22 
|-
| <math>\Gamma_k</math> 
| style="background:#FFFFCC" | '''Z''' 
| style="background:#FFCCFF" | '''Z'''<sub>2</sub> 
| style="background:#CCFFFF" | '''Z'''<sub>2</sub> 
| style="background:#ddf" | '''Z'''<sub>24</sub> 
| style="background:#FFFFCC" | 0 
| style="background:#FFCCFF" | 0 
| style="background:#CCFFFF" | '''Z'''<sub>2</sub> 
| style="background:#ddf" | '''Z'''<sub>240</sub> 
| style="background:#FFFFCC" | '''Z'''<sub>2</sub><sup>2</sup> 
| '''Z'''<sub>2</sub><sup>3</sup> 
| style="background:#CCFFFF" | '''Z'''<sub>6</sub> 
| style="background:#ddf" | '''Z'''<sub>504</sub> 
| style="background:#FFFFCC" | 0 
| '''Z'''<sub>3</sub> 
| style="background:#CCFFFF" | '''Z'''<sub>2</sub><sup>2</sup> 
| style="background:#ddf" | '''Z'''<sub>480</sub>&#x2295;'''Z'''<sub>2</sub> 
| style="background:#FFFFCC" | '''Z'''<sub>2</sub><sup>2</sup> 
| '''Z'''<sub>2</sub><sup>4</sup> 
| '''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>2</sub> 
| style="background:#ddf" | '''Z'''<sub>264</sub>&#x2295;'''Z'''<sub>2</sub> 
| '''Z'''<sub>24</sub> 
| '''Z'''<sub>2</sub><sup>2</sup> 
| '''Z'''<sub>2</sub><sup>2</sup> 
|}
Từ <math>k=23</math>, <math>\Gamma_k</math> trở nên phức tạp, ví dụ:

: <math>\Gamma_{23}=\Z_{65520}\oplus\Z_{24}\oplus\Z_{2}</math>
: <math>\Gamma_{23}=\Z_{16}\oplus\Z_8\oplus\Z_2\oplus\Z_9\oplus\Z_3\oplus\Z_5\oplus\Z_7\oplus\Z_{13}</math>

{| class="wikitable" style="text-align: center; margin: auto;"
|+''' Nhóm đồng luân ổn định ''' <math>\ \Gamma_k=\pi_{2k+2}(\mathbb S^{k+2})</math> với <math>k</math> nhỏ hơn 60
|<math>k</math>
|0
|1
|2
|3
|4
|5
|6
|7
|-
|<math>\Gamma_k</math>
| style="background:#FFFFCC" |'''Z'''
| style="background:#FFCCFF" |'''Z'''<sub>2</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>24</sub>='''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>3</sub>
| style="background:#FFFFCC" |0
| style="background:#FFCCFF" |0
| style="background:#CCFFFF" |'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>240</sub><br /><br />='''Z'''<sub>16</sub>&#x2295;'''Z'''<sub>3</sub>&#x2295;'''Z'''<sub>5</sub>
|- style="background: #eee"
|<math>k</math>
|8
|9
|10
|11
|12
|13
|14
|15
|-
|<math>\Gamma_k</math>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
|'''Z'''<sub>2</sub><sup>3</sup>
| style="background:#CCFFFF" |'''Z'''<sub>6</sub>='''Z'''<sub>2</sub>&#x2295;'''Z'''<sub>3</sub>
| style="background:#ddf" |'''Z'''<sub>504</sub><br /><br />='''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>9</sub>&#x2295;'''Z'''<sub>7</sub>
| style="background:#FFFFCC" |0
|'''Z'''<sub>3</sub>
| style="background:#CCFFFF" |'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#ddf" |'''Z'''<sub>480</sub>&#x2295;'''Z'''<sub>2</sub><br /><br />='''Z'''<sub>32</sub>&#x2295;'''Z'''<sub>2</sub>&#x2295;'''Z'''<sub>3</sub>&#x2295;'''Z'''<sub>5</sub>
|- style="background: #eee"
|<math>k</math>
|16
|17
|18
|19
|20
|21
|22
|23
|-
|<math>\Gamma_k</math>
| style="background:#FFFFCC" |'''Z'''<sub>2</sub><sup>2</sup>
|'''Z'''<sub>2</sub><sup>4</sup>
|'''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>2</sub>
| style="background:#ddf" |'''Z'''<sub>264</sub>&#x2295;'''Z'''<sub>2</sub><br /><br />='''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>2</sub>&#x2295;'''Z'''<sub>3</sub>&#x2295;'''Z'''<sub>11</sub>
|'''Z'''<sub>24</sub>
|'''Z'''<sub>2</sub><sup>2</sup>
|'''Z'''<sub>2</sub><sup>2</sup>
| style="background:#ddf" |'''Z'''<sub>16</sub>&#x2295;'''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>2</sub> &#x2295;'''Z'''<sub>9</sub>&#x2295;'''Z'''<sub>3</sub><br /><br /> &#x2295;'''Z'''<sub>5</sub>&#x2295;'''Z'''<sub>7</sub>&#x2295;'''Z'''<sub>13</sub>
|- style="background: #eee"
|<math>k</math>
|24
|25
|26
|27
|28
|29
|30
|31
|-
|<math>\Gamma_k</math>
|'''Z'''<sub>2</sub><sup>2</sup>
|'''Z'''<sub>2</sub><sup>2</sup>
|'''Z'''<sub>2</sub><sup>2</sup>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>24</sub>='''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>2</sub>
|'''Z'''<sub>3</sub>
|'''Z'''<sub>6</sub>='''Z'''<sub>2</sub>&#x2295;'''Z'''<sub>3</sub>
|
'''Z'''<sub>64</sub>&#x2295;'''Z'''<sub>2</sub><sup>2</sup>&#x2295;'''Z'''<sub>3</sub> &#x2295;'''Z'''<sub>5</sub>&#x2295;'''Z'''<sub>17</sub>
|- style="background: #eee"
|<math>k</math>
|32
|33
|34
|35
|36
|37
|38
|39
|-
|<math>\Gamma_k</math>
|'''Z'''<sub>2</sub><sup>4</sup>
|'''Z'''<sub>2</sub><sup>5</sup>
|'''Z'''<sub>4</sub>&#x2295;'''Z'''<sub>2</sub><sup>3</sup>
|'''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>2</sub><sup>2</sup>&#x2295;'''Z'''<sub>27</sub><br /><br />&#x2295;'''Z'''<sub>7</sub>&#x2295;'''Z'''<sub>19</sub>
|'''Z'''<sub>6</sub>='''Z'''<sub>2</sub>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>2</sub><sup>2</sup>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>2</sub>&#x2295;'''Z'''<sub>60</sub>=<br /><br />'''Z'''<sub>2</sub>&#x2295;'''Z'''<sub>4</sub>&#x2295;'''Z'''<sub>3</sub>&#x2295;'''Z'''<sub>5</sub>
|
'''Z'''<sub>16</sub>&#x2295;'''Z'''<sub>2</sub><sup>5</sup>&#x2295;'''Z'''<sub>3</sub><sup>2</sup>&#x2295;'''Z'''<sub>25</sub>&#x2295;'''Z'''<sub>11</sub>
|- style="background: #eee"
|<math>k</math>
|40
|41
|42
|43
|44
|45
|46
|47
|-
|<math>\Gamma_k</math>
|'''Z'''<sub>2</sub><sup>5</sup>&#x2295;'''Z'''<sub>4</sub>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>2</sub><sup>5</sup>
|'''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>2</sub><sup>2</sup>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>552</sub><br /><br />='''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>3</sub>&#x2295;'''Z'''<sub>23</sub>
|'''Z'''<sub>8</sub>
|'''Z'''<sub>16</sub>&#x2295;'''Z'''<sub>2</sub><sup>3</sup><br /><br /> &#x2295;'''Z'''<sub>9</sub>&#x2295;'''Z'''<sub>5</sub>
|'''Z'''<sub>2</sub><sup>4</sup>&#x2295;'''Z'''<sub>3</sub>
|
'''Z'''<sub>32</sub>&#x2295;'''Z'''<sub>4</sub>&#x2295;'''Z'''<sub>2</sub><sup>3</sup> &#x2295;'''Z'''<sub>9</sub>&#x2295;'''Z'''<sub>3</sub> &#x2295;'''Z'''<sub>5</sub>&#x2295;'''Z'''<sub>7</sub>&#x2295;'''Z'''<sub>13</sub>
|- style="background: #eee"
|<math>k</math>
|48
|49
|50
|51
|52
|53
|54
|55
|-
|<math>\Gamma_k</math>
|'''Z'''<sub>2</sub><sup>4</sup>&#x2295;'''Z'''<sub>4</sub>
|'''Z'''<sub>2</sub><sup>2</sup>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>3</sub>&#x2295;'''Z'''<sub>2</sub><sup>3</sup>
|'''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>4</sub>&#x2295;'''Z'''<sub>2</sub><sup>2</sup>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>2</sub><sup>3</sup>&#x2295;'''Z'''<sub>3</sub>
|'''Z'''<sub>2</sub><sup>4</sup>
|'''Z'''<sub>4</sub>&#x2295;'''Z'''<sub>2</sub>
|'''Z'''<sub>16</sub>&#x2295;'''Z'''<sub>3</sub><sup>2</sup>&#x2295;'''Z'''<sub>5</sub>&#x2295;'''Z'''<sub>29</sub>
|- style="background: #eee"
|<math>k</math>
|56
|57
|58
|59
|60
|61
|62
|63
|-
|<math>\Gamma_k</math>
|'''Z'''<sub>2</sub><sup>2</sup>
|'''Z'''<sub>2</sub><sup>4</sup>
|'''Z'''<sub>2</sub><sup>2</sup>
|'''Z'''<sub>8</sub>&#x2295;'''Z'''<sub>2</sub><sup>2</sup>&#x2295;'''Z'''<sub>9</sub><br /><br />&#x2295;'''Z'''<sub>7</sub>&#x2295;'''Z'''<sub>11</sub>&#x2295;'''Z'''<sub>31</sub>
|'''Z'''<sub>4</sub>
|
|
|
|}

=== Các nhóm đồng luân không ổn định ===
Một số nhóm đồng luân không ổn định:

* Với chiều 2 và 3 (<math>\pi_k(\mathbb S^2)=\pi_k(\mathbb S^3)</math>): 
** <math>\pi_3(\mathbb S^2)=\pi_3(\mathbb S^3)=\Z</math> 
** <math>\pi_5(\mathbb S^2)=\pi_5(\mathbb S^3)=\pi_4(\mathbb S^2)=\Z/(2)</math> 
** <math>\pi_6(\mathbb S^2)=\pi_6(\mathbb S^3)=\Z/(12)</math> 
* Với chiều 4: <math>\pi_7(\mathbb S^4)=\Z/(12)\oplus\Z</math>

=== Nhóm đồng luân vô hạn ===
Các nhóm đồng luân ổn định <math>\pi_{n+k}(\mathbb S^n)</math> là hữu hạn ngoại trừ <math>k=0</math> ((<math>\Gamma_0=\Z</math>).

Các nhóm đồng luân không ổn định là hữu hạn ngoại trừ các nhóm <math>\pi_{4p-1}(\mathbb S^{2p})</math> (với ''p'' > 0). Những nhóm này <math>\pi_3(\mathbb S^2)</math>, <math>\pi_7(\mathbb S^4)</math>, <math>\pi_{11}(\mathbb S^6)</math>,...) đẳng cấu với tổng trực tiếp của <math>\Z</math> và một nhóm hữu hạn.

==Tham khảo==
{{tham khảo}}

* Boris Doubrovine (de), Anatoli Fomenko et Sergueï Novikov, ''Géométrie contemporaine - Méthodes et applications'', tomes 2 et 3
* Claude Godbillon, ''Éléments de topologie algébrique''
* Fabien Morel, «''Groupes d'homotopie de sphères algébriques et formes quadratiques''», trong ''Leçons de mathématiques d'aujourd'hui'', vol. 3, Cassini, 2007

[[Thể loại:Tô pô đại số]]