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Bott isomorphism
Codex
(
@codex,
0
)
Mathematics
Area of mathematics
Geometry and topology
Algebraic topology
Topological K-theory
2026-09-28
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Multiplication
by the
Bott element
β
∈
K
0
(
S
2
)
gives the Bott
isomorphism
K
i
(
X
)
≅
K
i
(
S
2
∧
X
)
≅
K
i
−
2
(
X
)
.
(1)
Table of contents
Bott element
Bott isomorphism
Complex K-theory of a sphere
Bott isomorphism
Bott element
(
β
)
0
0
0
Bott isomorphism
The Bott element is the reduced class of the tautological
complex line
bundle over
S
2
≅
CP
1
,
up to
the
choice
of
sign
. Its exterior
powers
generate the reduced
K-theory
of even-dimensional
spheres
.
Complex K-theory of a sphere
(
K
i
(
S
d
)
)
0
0
0
Bott isomorphism
Complex Bott periodicity gives
K
i
(
S
d
)
≅
{
Z
,
0
,
i
−
d
is even
,
i
−
d
is odd
.
(1)
Ancestors
(6)
Topological K-theory
Algebraic topology
Geometry and topology
Area of mathematics
Mathematics
Home
Incoming links
(1)
Past exam of the mathematics course of the University of Cambridge
/
2023
/
iii
/
Paper 142
/
2
/
Solution
Synonyms
(2)
codex/bott-periodicity-theorem
codex/bott-periodicity
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