The cobordism of involutions on orientable manifolds

This note calculates the cobordism of smooth, oriented manifolds with orientation-reversing involution, and also the 2-torsion in the cobordism of such manifolds with orientationpreserving involution. In particular, it is shown that the forgetful homomorphism maps both groups monomorphically into the cobordism of unoriented manifolds with involution. 1. Statement of results. Let 0* be the cobordism theory of smooth oriented manifolds M with involution T; we do not require T to be orientation preserving. Then €if = &~¡f®G'X^ where the equivalence class [M, T] is in 0* (respectively, 0%), if Fis orientation reversing (respectively, preserving). The notation agrees with that of [4]. In this note we determine fí>*©torsion &%■ Rosenzweig [5] proved that the torsion of <$% is all of order 2. In 0* every class is of order 2, for if T is orientation reversing, (Mu MJuT)^ d(M x I,T x Id) under the diffeomorphism IduF. &X was also studied by Conner [2], and 0 has recently been studied by Lee and Wasserman [4], who show how classes of 6^ are detected by equivariant characteristic numbers. Our results complement these two papers, and we offer a short proof (Lemma 4) of a result which appears in both. Theorem 1. Let I% be the cobordism of unoriented manifolds with involution [3, Chapter IV]. The forgetful map p: (9^-^-1^ is monic on 2t ors ion. From [6] it follows that the torsion is mapped monomorphically into the Wall bordism group W*(\/JLX TBO(j)) (see [7, Chapter VIII]). An element of this group is represented by a vector bundle v—>F, whose disk bundle Dv is a Wall manifold [8], together with a classifying map Received by the editors November 14, 1972. AMS (MOS) subject classifications (1970). Primary 57D85; Secondary 57D85.