Surgery of closed manifolds with dihedral fundamental group

Abstract

ABSTRACT. In the paper the obstruction groups to obtaining simple homotopy equivalence by surgery from normal degree 1 maps of closed manifolds with dihedral fundamental group are computed. The cases of trivial orientation for the dihedral group and nontrivial orientation for the order 2 cyclic subgroup are considered. New results concerning the Browder-Livesey groups and natural maps of L-groups arising in index 2 inclusions of the cyclic group into the dihedral group are obtained.

Arf invariants of codimension one in a Wall group of the dihedral group

“…Statement (b) follows from Theorem 5 in [20]. Indeed, it implies that the elements realisable by closed submanifolds form a subgroup of index 4 of L 3 (D + ).…”

“…Therefore, the diagram (3.7) takes the following form: Proof. Statement (a) is proved using the isomorphism from [20], p. 247, the calculations of maps in [22], Case C, and the above results on the Wall and Browder-Livesay groups, since the second line of the diagram (3.8) can be written as…”

“…The notation D 3 is due to the fact that this group is included in an infinite series of 2-groups; see [20]. We consider the following subgroups of D:…”

“…where D + = D + a× ȧ, the orientation character is trivial on I a× ȧ, and b ∈ D + \ I a× ȧ. The groups and maps involved in (3.7) are known from [19], [20], [17], [21], p. 527, and [22]. Since the notation in these works differs from the notation introduced above, we describe the relationship between the Wall and Browder-Livesay groups using two different systems of notation.…”

“…Since the notation in these works differs from the notation introduced above, we describe the relationship between the Wall and Browder-Livesay groups using two different systems of notation. In the notation used by Wall and in [19], [20] and [22] we have…”