A Borsuk-Ulam theorem for ( Z p ) k -actions on products of (mod p ) homology spheres
Abstract
It is proved that for a product action of
(
Z
p
)
k
on a product of (mod p) homology spheres
N
n
1
×...×
N
n
k
, where all
n
i
's are assumed to be odd if
p
is odd, and any continuous map
f:
N
n
1
×...×
N
n
k
→
R
m
the set
A(f)={x∈
N
n
1
×...×
N
n
k
|f(x)=f(gx)∀g∈(
Z
p
)
k
}
has dimension at least
n
1
+...+
n
k
−m(
p
k
−1)
, provided
n
i
≥m
p
i−1
(p−1)
for all
i(1≤i≤k)
. Moreover, if
n
i
≥m
p
k−1
(p−1)
for all
i(1≤i≤k)
then the free action
μ
can be assumed arbitrary.