Bordism Theory Research Articles

Multiplications in the Complex Bordism Theory with Singularities

Multiplications in the Complex Bordism Theory with Singularities

Stiefel-Whitney Homology Classes and Bordism

We develop the theory of $\bmod 2$ Stiefel-Whitney homology classes for Euler polyhedra. We then describe a simple method of obtaining p.1. bordism theories. Finally, we define the ungraded bordism theory of Euler spaces and show that it is isomorphic to ordinary total homology.

Bordism of manifolds with oriented boundaries

A bordism theory is defined for manifolds with oriented boundaries. The relation of this theory with the ordinary bordism theories is shown. These bordism classes are then characterized via characteristic numbers.

Bordismengruppen unit�rer Torusmannigfaltigkeiten

Let G be a k-dimensional torus. Let U * G , denote the homotopical unitary bordism theory. We show that restriction to the fixed point set determines an element of U * G . This implies that the bordism class of a unitary G-manifold is determined by its characteristic numbers in equivariant K-theory.

On Bordism Theory of Manifolds with Singularities.

On Bordism Theory of Manifolds with Singularities.

The Todd character and cohomology operations

In “On the Conflict of Bordism of Finite Complexes” [ J. Differential Geometry], Conner and Smith introduced a homomorphism called the Todd character, relating complex bordism theory to rational homology. Specifically the Todd character consists of a family of homomorphisms th r: MU s(X) → H s→r(X; Q) . In L. Smith, The Todd character and the integrality theorem for the Chern character, Ill. J. Math. it was shown (note that the indexing of the Todd character is somewhat different here) that there was an integrality theorem for th analogous to the Adams integrality theorem for the Chern character J. F. Adams, On the Chern character and the structure of the unitary group, Proc. Cambridge Philos. Soc. 57 (1961), 189–199; On the Chern character revisted, Ill. J. Math. Now Adams' first paper contains a wealth of information about the Chern character in addition to the integrality theorem already mentioned. Our objective in the present note is to derive analogous results for the Todd character. As in Smith these may then be used to deduce the results of Adams for the Chern character.

Oriented and Weakly Complex Bordism Algebra of Free Periodic Maps

Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly complex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps.

Oriented and weakly complex bordism algebra of free periodic maps

Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly complex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps.

Equivariant cobordism and duality

We consider equivariant cobordism theory, defined by means of an equivariant Thorn spectrum; in particular, we investigate the relationship between this theory and the more geometric equivariant bordism theory, showing that there is a Poincaré-Lefschetz duality theorem which is valid in this setting.

BORDISMS WITH THE ACTION OF THE GROUPZpAND FIXED POINTS

A full description of the ring of unitary bordisms with the action of Zp and of the ring of all admissible collections of fixed submanifolds under the action of Zp is given in terms of generators and relations. The calculations are based on a special choice of polynomial generators in the ordinary ring of unitary bordisms and Poincare duality in bordism theory. Bibliography: 3 titles.

Differentiable transformation groups on homotopy spheres.

Abstract : Techniques and results of differentiable topology are employed in the study of differentiable transformation groups. This paper differs from recent works with the same objective in that pasting techniques rather than bordism theory are used. (Author)

Duality and orientability in bordism theories

Duality and orientability in bordism theories

Kunneth Formulas for Bordism Theories

Kunneth Formulas for Bordism Theories

Orientability of manifolds for generalised homology theories

Recently, generalised homology theories have received much attention ([5], [6], [19] etc.). An interesting question which arises in these theories is the orientability of manifolds [20]; as Whitehead remarks, this is a delicate question. In this paper we shall show that if there exists a microbundle 4 over a closed manifold M such that 4 is A-orientable in the sense of Dold [6], and such that the top homology class of the Thom space M4 is spherical, then M is A-orientable in the sense of Whitehead [19]. Hence M satisfies Poincare duality with coefficient spectrum A. Let N be a compact smooth manifold whose boundary ON= Y is a homotopy sphere which bounds a 7r-manifold: form a manifold M by adjoining to N a cone on its boundary S. We shall call such manifold-s M almost-smooth: the manifolds of Kervaire [11], Smale [16], Eells and Kuiper [8] (see also [18]), which do not have the same homotopy type as any closed smooth manifold, are all almost smooth. We shall show that, over any such M, there is a vector bundle 4 such that the top homology class of its Thom complex MX is spherical. It follows that (provided M is orientable in the usual sense), M is orientable for K-theory and KO-theory [2] and for bordism theory [1]. It also follows from the existence of such bundles that the hypothesis on the signature in the theorem of Browder [4], which gives necessary and sufficient conditions for a finite, simply-connected CW-complex to have the homotopy type of a closed smooth manifold (of dimension : 3,4), cannot be dispensed with. Microbundles and Thom spaces. We consider pairs (i,j) of maps, with ji the identity map of some fixed space B. Two pairs (i1,jl) and (i2,j2) are equivalent if there is a third pair (i3,j3) and a commutative diagram,

The 𝑆𝑈-bordism theory

The 𝑆𝑈-bordism theory

A bordism theory for actions of an abelian group

A bordism theory for actions of an abelian group