Bott periodicity theorem
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
:
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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, "![]() |
[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 ![]() |
[a2] | M. Karoubi, "![]() |
Bott periodicity theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bott_periodicity_theorem&oldid=24047