Maps which induce the zero map on homotopy

Kahn, Donald W.

doi:10.2140/pjm.1965.15.537

“…To a pair of spaces Emery Thomas [8] has defined invariants whose vanishing implies that a map between these spaces is null-homotopic if all of its induced cohomology homomorphisms are zero. To a pair of spaces Donald Kahn [1] has assigned groups whose vanishing implies that a map between these spaces is null-homotopic if all of its induced homotopy homomorphisms are zero. For these classification problems we will give necessary and sufficient conditions from which the above sufficiency results will follow.…”

Section: Introductionmentioning

confidence: 99%

Necessary and sufficient conditions for homotopy classification by cohomology and homotopy homomorphisms

Sternstein¹

1972

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…To a pair of spaces Emery Thomas [8] has defined invariants whose vanishing implies that a map between these spaces is null-homotopic if all of its induced cohomology homomorphisms are zero. To a pair of spaces Donald Kahn [1] has assigned groups whose vanishing implies that a map between these spaces is null-homotopic if all of its induced homotopy homomorphisms are zero. For these classification problems we will give necessary and sufficient conditions from which the above sufficiency results will follow.…”

Section: Introductionmentioning

confidence: 99%

“…An example is found letting X=S5 and Y=(SzVJfes)\J2ge6 where/generates iri(S3)=Z2 and g generates ■n-i(S3Kjfei)=Z. Donald Kahn [1] has assigned groups whose vanishing implies that a map is null-homotopic if all of its induced homotopy homomorphisms are zero. The vanishing of these groups implies that our invariants, [ker(^+2| X) r\ ker ^Aj&(*,*), vanish; see [7].…”

Section: Introductionmentioning

confidence: 99%

Necessary and Sufficient Conditions for Homotopy Classification by Cohomology and Homotopy Homomorphisms

Sternstein¹

1972

Proceedings of the American Mathematical Society

0

View full text Add to dashboard Cite

Abstract.Using obstruction theory tools to a pair of spaces two invariants are defined whose vanishing is a necessary and sufficient condition for homotopy classification of certain maps between these spaces by the induced cohomology and homotopy homomorphisms respectively. Introduction.By the Hopf Theorem the set of homotopy classes of maps from a complex of dimension n into an «-sphere can be classified by the cohomology homomorphism each class induces. To a pair of spaces Emery Thomas [8] has defined invariants whose vanishing implies that a map between these spaces is null-homotopic if all of its induced cohomology homomorphisms are zero. To a pair of spaces Donald Kahn [1] has assigned groups whose vanishing implies that a map between these spaces is null-homotopic if all of its induced homotopy homomorphisms are zero. For these classification problems we will give necessary and sufficient conditions from which the above sufficiency results will follow.

show abstract

“…(X) for i £ n . D. W. Kahn [3] showed that for every unitary bundle £ over X , the nth Chern class c (£ ) is contained in a 2 subgroup of H (X, Z) which belongs to •£ (X, 2n-l) . If X(n) denotes the order of the group TT (X) , we shall show further that 2n \(2n-l)c (£) is contained in a subgroup of H (X, Z) which belongs to S (X, 2n-2) .…”

mentioning

confidence: 99%