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Hatcher-Algebraic-Topology
· Problem 2.2.2
Homology
Mathematics — Graph Theory · degree-theory
Problem
Given a map
f
:
S
2
n
→
S
2
n
f: S^{2n} \to S^{2n}
f
:
S
2
n
→
S
2
n
, show that there is some point
x
∈
S
2
n
x \in S^{2n}
x
∈
S
2
n
with either
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
or
f
(
x
)
=
−
x
f(x) = -x
f
(
x
)
=
−
x
. Deduce that every map
R
P
2
n
→
R
P
2
n
\mathbb{RP}^{2n} \to \mathbb{RP}^{2n}
RP
2
n
→
RP
2
n
has a fixed point. Construct maps
R
P
2
n
−
1
→
R
P
2
n
−
1
\mathbb{RP}^{2n-1} \to \mathbb{RP}^{2n-1}
RP
2
n
−
1
→
RP
2
n
−
1
without fixed points from linear transformations
R
2
n
→
R
2
n
\mathbb{R}^{2n} \to \mathbb{R}^{2n}
R
2
n
→
R
2
n
without eigenvectors.
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