On Wall Manifolds with Almost Free $\mathbf{Z}_{2^k}$ Actions

Abstract

In order to understand the bordism classification of finite group actions on oriented manifolds, it is useful to consider some notion of manifolds with equivariant Wall structures.In [8], C. Kosniowski and E. Ossa studied the bordism theory $W_{*}$ ( $Z_{2}$ ; All) of Wall manifolds with unrestricted involutions and determined completely the bordism theory $\Omega_{*}$ ( $Z_{2}$ ; All) of oriented involutions, especially its torsion part as the image of the Bockstein homomorphism $\beta:W_{*}(Z_{2};All)\rightarrow\Omega_{*}$ ( $Z_{2}$ ; All).In this paper, we treat an almost free $Z_{2^{k}}$ action on Wall manifold, i.e., one for which only the $Z_{2}\subset Z_{2^{k}}$ may possibly fix points on manifold.From the viewpoint of action, such object is exactly Wall manifold with action of type $(Z_{2^{k}}, 1)$ in [13].In section 1, we study the bordism theory $W_{*}(Z_{2^{k}};Af)$ of these objects.By the map which ignores Wall structures, the theories $W_{*}$ ( $Z_{2^{k}}$ ; Free) and $W_{*}$ ( $Z_{2^{k}};Af$ , Free) are derived from the corresponding unoriented theories as usual (Propositions 1.4 and 1.8).In particular, we have that $W_{*}$ ( $Z_{2^{k}};Af$ , Free) is the sum of three parts; the images $Im(t)$ of two kinds of extensions from $Z_{2}$ actions and another part $L_{*}$ .Using these results, we obtain the exact sequence for the triple ($Af$, Free, $\emptyset$ ) (Proposition 1.11), and the $W_{*}$ -module structure of $W_{*}(Z_{21c};Af)$ (Theorem 1.19).There the classes $\{V(O, 2n+2)\}$ (Definition 1.17) are useful to describe the part $K_{t}$ which lies in $Im(t)\subset W_{*}$ ( $Z_{2^{k}};Af$ , Free), while the part $L_{*}$ is isomorphic to $L_{*}$ naturally.In section 2, we describe the image $\mathcal{T}$ of the map $\beta:W_{*}(Z_{2^{k}};Af)\rightarrow\Omega_{*}(Z_{2^{lc}};Af)$ ;the bordism module of orientation preserving almost free $Z_{2^{k}}$ actions, and describe the torsion part of order 2 (Theorem 2.3).As an application, we study the image of $I_{*}:$ $\Omega_{*}(Z_{4};Free)\rightarrow\Omega_{*}(Z_{4};Af)$ ; the forgetful homomorphism by using the result of principal $Z_{2^{k}}$ actions in [5] (Theorem 2.9).The author would like to thank the referee for his many valuable comments.