View source for Bott periodicity theorem

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.

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" />:

<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>

====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, "<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>



====Comments====


====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>