交错群A_4在同伦S^2×S^2上的作用

On spin alternating group actions on homotopy S^2×S^2

设X是一个与S2×S2同伦等价的闭光滑四维流形,本文利用Seiberg-Witten理论证明了如果X上有一个交错群A4的Spin作用使得b2+(X/A4)=b2+(X),则X的等变Dirac算子的指标满足IndA4DX=a(1+ξ+ξ2-η),其中a是一个整数,1,ξ,ξ2和η分别为A4的度数为1,1,1,和3的4个不可约特征标.

Let X be a closed smooth 4-manifold which is homotopy equivalent to S2 × S2. In this paper we use Seiberg-Witten theory to prove that if X admits a spin alternating A4 action with b2^＋ （X/A4 ） = b2^＋ （X）, then as an element of R（A4） ,IndA4Dx = a （1 ＋ ξ ＋ ξ^2 -η）, where a is an integer, 1 ,ξ,ξ^2 and η are 4 irreducible characters of A4 of degree 1,1,1, and 3 respectively.