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Lefschetz-Hopf fixed-point theorem
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Mathematics
Area of mathematics
Geometry and topology
Algebraic topology
Lefschetz fixed-point theorem
Created
2026-09-24
Updated
2026-09-24
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If
a
smooth self-
map
of
a
compact
manifold
has only isolated nondegenerate fixed points, then
L
(
f
)
=
∑
f
(
x
)
=
x
sign
det
(
I
−
D
f
x
)
.
(1)
Every summand has
absolute value
one, so the
number
of fixed points is at least
∣
L
(
f
)
∣
.
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(6)
Lefschetz fixed-point theorem
Algebraic topology
Geometry and topology
Area of mathematics
Mathematics
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(1)
Past exam of the mathematics course of the University of Cambridge
/
2026
/
iii
/
Paper 114
/
4
/
Solution
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