论文标题
borsuk-环状$ p $ - groups
A Borsuk--Ulam theorem for cyclic $p$-groups
论文作者
论文摘要
我们描述了一个结缔组织$ k $ - 理论 - 乌拉姆/布尔金 - Yang定理，用于循环组的Prime $ p $的电源。考虑两个有限的维度复杂表示$ u $和$ v $的循环组$ z /p^{k+1} $的订单$ p^{k+1} $，其中$ k \ geq 0 $。对于$ 0 \ leq l \ leq k $，我们为$ v $的子空间编写$ v_l $，由订单$ p^l $的循环子组修复，并要求固定子空间，$ v_ {k+1} $，为零，$ v_k $是非零的。 pur $δ（v）= \ sum_ {l = 0}^k p^l dim_c（v_l/v_ {l+1}） - （p^k-1）$。然后，从$ u $（某些不变的内部产品）中的任何$ z /p^{k+1} $ - map $ s（u）\ to v $的零集覆盖尺寸大于或等于$ 2（dim_c u-δ（v）-1）-1 -1）
We describe a connective $K$-theory Borsuk--Ulam/Bourgin--Yang theorem for cyclic groups of order a power of a prime $p$. Consider two finite dimensional complex representations $U$ and $V$ of the cyclic group $Z /p^{k+1}$ of order $p^{k+1}$, where $k\geq 0$. For $0\leq l\leq k$, we write $V_l$ for the subspace of $V$ fixed by the cyclic subgroup of order $p^l$, and require that the fixed subspace, $V_{k+1}$, be zero and that $V_k$ be non-zero. Put $δ(V)=\sum_{l=0}^k p^l dim_C (V_l/V_{l+1})-(p^k-1)$. Then the zero-set of any $Z /p^{k+1}$-map $S(U) \to V$ from the unit sphere in $U$ (for some invariant inner product) has covering dimension greater than or equal to $2(dim_C U - δ(V)-1)$, if $dim_C U> δ(V)$.
