In (10), (11) Wall determined the structure of the cobordism ring introduced by Thom in (9). Among Wall's results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin(5), (6) and Dold(3) which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are ‘cohomology-type’ exact sequences arising in the well-known way from mappings into a universal space. The appropriate ‘cohomology’ theory is constructed by taking as universal space the Thom complex MSO ( n ), for n large. This gives rise to (oriented) cobordism groups MSO *( X ) of a space X .
[1]
N. Steenrod.
Topology of Fibre Bundles
,
1951
.
[2]
N. Steenrod.
The Topology of Fibre Bundles. (PMS-14)
,
1951
.
[3]
Jean-Pierre Serre,et al.
Cohomologie modulo 2 des complexes d’Eilenberg-MacLane
,
1953
.
[4]
R. Thom.
Quelques propriétés globales des variétés différentiables
,
1954
.
[5]
Note on the cobordism ring
,
1959
.
[6]
M. Atiyah,et al.
Riemann-Roch theorems for differentiable manifolds
,
1959
.
[7]
C. Wall.
Determination of the Cobordism Ring
,
1960
.
[8]
J. Milnor.
On the Cobordism Ring Ω ∗ and a Complex Analogue, Part I
,
1960
.