Difference between revisions of "Bott periodicity theorem"
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 17:31, 31 March 2012
A fundamental theorem in 
-theory which, in its simplest form, states that for any (compact) space 
there exists an isomorphism between the rings 
and 
. More generally, if 
is a complex vector bundle over 
and 
is the projectivization of 
, then the ring 
is a 
-algebra with one generator 
and a unique relation 
, where 
is the image of a vector bundle 
in 
and 
is the Hopf fibration over 
. This fact is equivalent to the existence of a Thom isomorphism in 
-theory for complex vector bundles. In particular, 
. Bott's periodicity theorem was first demonstrated by R. Bott [1] using Morse theory, and was then re-formulated in terms of 
-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 
, consisting in the fact that 
, where 
is the space of loops on 
, and 
is weak homotopy equivalence, in particular 
for 
where 
is the 
-th homotopy group. Similarly, for the orthogonal group 
:
 |
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, " -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  " , Benjamin (1969) MR0258020 Zbl 0194.23904 |
| [a2] | M. Karoubi, " -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=18004
Bott periodicity theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bott_periodicity_theorem&oldid=18004
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