Thom Spaces Research Articles

Thom modules and mod 𝑝 spherical fibrations

In this paper we show that every finite Thom module over the ring of invariants of a finite nonmodular group can be realized as mod p p cohomology of the Thom space of a spherical fibration.

Thom modules and pseudoreflection groups

This paper studies the structure of Thom modules. These are the purely algebraic analogue of the cohomology of a Thom space. Special emphasis is placed on Thom modules over the ring of invariants of a finite group generated by pseudoreflections.

Spanier-Whitehead duality in étale homotopy

We construct a ( mod - l ) (\bmod {\text {-}}l) Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a ( mod - l ) (\bmod {\text {-}}l) Spanier-Whitehead dual.

The rational homotopy of thom spaces and the smoothing of homology classes

The rational homotopy of thom spaces and the smoothing of homology classes

The rational homotopy of Thom spaces and the smoothing of isolated singularities

The rational homotopy of Thom spaces and the smoothing of isolated singularities

A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )

A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )

On immersions of n-Manifolds

A classical conjecture in differential topology is that an n -manifold immerses in R 2 n - α ( n ) , where α ( n ) is the number of ones in the dyadic expansion of n . We prove a corollary of this conjecture, namely, that at the Thom space level, the map T(v) → MO factors through MO ( n - α ( n )).

Poincare Duality Cobordism

Consider the following problem: What is the relation between the group of oriented cobordism classes of oriented Poincare duality spaces (denoted QPD.) and the stable homotopy group wrMSG, where MSG is the Thom space which arises from the universal orientable spherical fibrations. In particular, the result of Spivak [8] allows the construction of a Pontrjagin-Thom map p: QPD.wr MSG. One therefore, wishes to know how close p is to being an isomorphism. Of course, it is not true that p is an isomorphism, for it is easily seen that r*MSG is a finite group in each dimension whereas, in dimensions 4k, the index of an oriented Poincare duality space defines a non-trivial homomorphism Q'-'D Z. Or, to put it another way, there exist two Poincare duality spaces M4k and N4k, with a degree one map f: M4k N4k covered by a map of Spivak normal bundles, (which in turn extends to a degree-one map of ambient spheres S4k+m S4k+m m large) yet with index M# index N. For example, the non-smoothable Milnor manifolds of dimension 4k have fiberhomotopy trivial normal bundles, (i.e. trivial normal fibrations when regarded as P.D. spaces) yet none of them have zero index, hence none is oriented cobordant to zero.

Pontryagin squares in the Thom space of a bundle

The object of this note is to determine the action of the Pontryagin squares in the cohomology of the Thorn space of a vector bundle. This computation is then applied to the case of the normal bundle of a manifold imbedded in Euclidean space to give simplified proofs of some theorems of Mahowald. The first of Mahowald's theorems [3] was inspired by some 1940 results of Whitney [9], who showed that in certain cases the Euler class (with twisted integer coefficients) of the normal bundle of a nonorientable surface imbedded in Euclidean 4-space could be nonzero. This contrasts with the well-known theorem that the Euler class of the normal bundle of an orientable manifold in Euclidean space is always zero.

The Cohomology of the Loop Spaces of Thom Spaces

The Cohomology of the Loop Spaces of Thom Spaces

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,