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A fundamental theorem in [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172201.png" />-theory]] which, in its simplest form, states that for any (compact) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172202.png" /> there exists an isomorphism between the rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172204.png" />. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172205.png" /> is a complex vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172207.png" /> is the projectivization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172208.png" />, then the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b0172209.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722010.png" />-algebra with one generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722011.png" /> and a unique relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722013.png" /> is the image of a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722016.png" /> is the Hopf fibration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722017.png" />. This fact is equivalent to the existence of a Thom isomorphism in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722018.png" />-theory for complex vector bundles. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722019.png" />. Bott's periodicity theorem was first demonstrated by R. Bott [[#References|[1]]] using Morse theory, and was then re-formulated in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722020.png" />-theory [[#References|[6]]]; an analogous theorem has also been demonstrated for real fibre bundles.
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Bott's periodicity theorem establishes the property of the stable homotopy type of the unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722021.png" />, consisting in the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722023.png" /> is the space of loops on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722025.png" /> is weak homotopy equivalence, in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722027.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722028.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722029.png" />-th homotopy group. Similarly, for the orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722030.png" />:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722031.png" /></td> </tr></table>
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A fundamental theorem in [[K-theory| $  K $-
 +
theory]] which, in its simplest form, states that for any (compact) space  $  X $
 +
there exists an isomorphism between the rings  $  K(X) \otimes K(S  ^ {2} ) $
 +
and  $  K(X \times S  ^ {2} ) $.
 +
More generally, if  $  L $
 +
is a complex vector bundle over  $  X $
 +
and  $  P(L \oplus 1) $
 +
is the projectivization of  $  L \oplus 1 $,
 +
then the ring  $  K(P(L \oplus 1)) $
 +
is a  $  K(X) $-
 +
algebra with one generator  $  [H] $
 +
and a unique relation  $  ([H] - [1])([L][H] - [1]) = 0 $,
 +
where  $  [E] $
 +
is the image of a vector bundle  $  E $
 +
in  $  K(X) $
 +
and  $  H  ^ {-1} $
 +
is the Hopf fibration over  $  P(L \oplus 1) $.
 +
This fact is equivalent to the existence of a Thom isomorphism in  $  K $-
 +
theory for complex vector bundles. In particular,  $  P(1 \oplus 1) = X \times S  ^ {2} $.  
 +
Bott's periodicity theorem was first demonstrated by R. Bott [[#References|[1]]] using Morse theory, and was then re-formulated in terms of  $  K $-
 +
theory [[#References|[6]]]; an analogous theorem has also been demonstrated for real fibre bundles.
  
====References====
+
Bott's periodicity theorem establishes the property of the stable homotopy type of the unitary group  $  U $,  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Bott, "The stable homotopy of the classical groups" ''Ann. of Math. (2)'' , '''70''' : 2 (1959) pp. 313–337 {{MR|0110104}} {{ZBL|0129.15601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) {{MR|0163331}} {{ZBL|0108.10401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722032.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.C. Moore, "On the periodicity theorem for complex vector bundles" , ''Sem. H. Cartan'' (1959–1960)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.F. Atiyah, R. Bott, "On the periodicity theorem for complex vector bundles" ''Acta Math.'' , '''112''' (1964) pp. 229–247 {{MR|0178470}} {{ZBL|0131.38201}} </TD></TR></table>
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consisting in the fact that  $  {\Omega  ^ {2} } U \sim U $,  
 +
where  $  \Omega X $
 +
is the space of loops on  $  X $,  
 +
and  $  \sim $
 +
is [[weak homotopy equivalence]], in particular  $  \pi _ {i} (U) = \pi _ {i+2} (U) $
 +
for  $  i = 0, 1 \dots $
 +
where  $  \pi _ {i} $
 +
is the  $  i $-
 +
th homotopy group. Similarly, for the orthogonal group  $  O $:
  
 +
$$
 +
\Omega  ^ {8}
 +
O  \sim  O,\ \
 +
\pi _ {i} (O)  =  \pi _ {i+ 8 }
 +
(O).
 +
$$
  
 +
====References====
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Bott, "The stable homotopy of the classical groups" ''Ann. of Math. (2)'' , '''70''' : 2 (1959) pp. 313–337 {{MR|0110104}} {{ZBL|0129.15601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) {{MR|0163331}} {{ZBL|0108.10401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.C. Moore, "On the periodicity theorem for complex vector bundles" , ''Sem. H. Cartan'' (1959–1960)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.F. Atiyah, R. Bott, "On the periodicity theorem for complex vector bundles" ''Acta Math.'' , '''112''' (1964) pp. 229–247 {{MR|0178470}} {{ZBL|0131.38201}} </TD></TR></table>
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Bott, "Lectures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722033.png" />" , Benjamin (1969) {{MR|0258020}} {{ZBL|0194.23904}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Karoubi, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017220/b01722034.png" />-theory" , Springer (1978) {{MR|0488029}} {{ZBL|0382.55002}} </TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Bott, "Lectures on $K(X)$" , Benjamin (1969) {{MR|0258020}} {{ZBL|0194.23904}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Karoubi, "$K$-theory" , Springer (1978) {{MR|0488029}} {{ZBL|0382.55002}} </TD></TR></table>

Latest revision as of 13:07, 24 December 2020


A fundamental theorem in $ K $- theory which, in its simplest form, states that for any (compact) space $ X $ there exists an isomorphism between the rings $ K(X) \otimes K(S ^ {2} ) $ and $ K(X \times S ^ {2} ) $. More generally, if $ L $ is a complex vector bundle over $ X $ and $ P(L \oplus 1) $ is the projectivization of $ L \oplus 1 $, then the ring $ K(P(L \oplus 1)) $ is a $ K(X) $- algebra with one generator $ [H] $ and a unique relation $ ([H] - [1])([L][H] - [1]) = 0 $, where $ [E] $ is the image of a vector bundle $ E $ in $ K(X) $ and $ H ^ {-1} $ is the Hopf fibration over $ P(L \oplus 1) $. This fact is equivalent to the existence of a Thom isomorphism in $ K $- theory for complex vector bundles. In particular, $ P(1 \oplus 1) = X \times S ^ {2} $. Bott's periodicity theorem was first demonstrated by R. Bott [1] using Morse theory, and was then re-formulated in terms of $ K $- theory [6]; an analogous theorem has also been demonstrated for real fibre bundles.

Bott's periodicity theorem establishes the property of the stable homotopy type of the unitary group $ U $, consisting in the fact that $ {\Omega ^ {2} } U \sim U $, where $ \Omega X $ is the space of loops on $ X $, and $ \sim $ is weak homotopy equivalence, in particular $ \pi _ {i} (U) = \pi _ {i+2} (U) $ for $ i = 0, 1 \dots $ where $ \pi _ {i} $ is the $ i $- th homotopy group. Similarly, for the orthogonal group $ O $:

$$ \Omega ^ {8} O \sim O,\ \ \pi _ {i} (O) = \pi _ {i+ 8 } (O). $$

References

[1] R. Bott, "The stable homotopy of the classical groups" Ann. of Math. (2) , 70 : 2 (1959) pp. 313–337 MR0110104 Zbl 0129.15601
[2] J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) MR0163331 Zbl 0108.10401
[3] M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967) MR224083
[4] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804
[5] J.C. Moore, "On the periodicity theorem for complex vector bundles" , Sem. H. Cartan (1959–1960)
[6] M.F. Atiyah, R. Bott, "On the periodicity theorem for complex vector bundles" Acta Math. , 112 (1964) pp. 229–247 MR0178470 Zbl 0131.38201

Comments

References

[a1] R. Bott, "Lectures on $K(X)$" , Benjamin (1969) MR0258020 Zbl 0194.23904
[a2] M. Karoubi, "$K$-theory" , Springer (1978) MR0488029 Zbl 0382.55002
How to Cite This Entry:
Bott periodicity theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bott_periodicity_theorem&oldid=24047
This article was adapted from an original article by A.F. Shchekut'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article