On the Order of Certain Elements of $J(X)$ and the Adams Conjecture

Kono, Akira

doi:10.2977/prims/1195185264

Cited by 5 publications

(4 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof to be presented below is, apart from minor rewriting in the language of fibrewise fixed-point theory, that given by Kono in [9], [10]. In fact the map ϑ, or rather a desuspension of that map, can be traced back to the work of James in the '50s; see [4] (2.11).…”

Section: The Brauer-segal Theoremmentioning

confidence: 99%

“…This result was used by Kono in [10] to establish a precise form of the (non-equivariant) Adams conjecture for complex vector bundles over a finite complex. To complete the proof one needs, as in the original topological proof due to Becker and Gottlieb, the non-trivial fact that the J-map K 0 (X ) → Sph 0 (X ) commutes with stable maps.…”

Section: The Brauer-segal Theoremmentioning

confidence: 99%

“…In [12] Segal proved that the evaluation map ε : {X + ; (BT) + } → K 0 (X ), defined by ε( f ) := f * (H ), where H ∈ K 0 (BT) is the Hopf line bundle over the classifying space BT of the circle group T, is a split surjection from the abelian group of stable maps X + → (BT) + to the complex K -group of the space X. His proof used the result of Brauer [3] that every element of the complex representation ring R(G) of a finite group G is a linear combination of classes induced from 1-dimensional representations of subgroups of G. An elegant new proof of Segal's theorem was given by Kono in [9], [10]. This proof naturally includes an equivariant extension of the theorem.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Brauer induction and equivariant stable homotopy

Crabb

2001

Arch. Math.

View full text Add to dashboard Cite

Brauer's induction theorem, published in 1951, asserts that every element of the complex representation ring R(G ) of a finite group G is a linear combination of classes induced from 1-dimensional representations of subgroups of G. In 1987, Snaith formulated an explicit version of the induction theorem. Using the methods of equivariant fibrewise stable homotopy theory, specifically fixed-point theory, this note clarifies the relation between the explicit Brauer induction theorem due to Snaith, Boltje and Symonds and a topological splitting theorem established by Segal in 1973. Introduction.Let X be a finite complex and X + the pointed space obtained by adjoining a disjoint base-point to X. In [12] Segal proved that the evaluation map ε :the Hopf line bundle over the classifying space BT of the circle group T, is a split surjection from the abelian group of stable maps X + → (BT) + to the complex K -group of the space X. His proof used the result of Brauer [3] that every element of the complex representation ring R(G) of a finite group G is a linear combination of classes induced from 1-dimensional representations of subgroups of G. An elegant new proof of Segal's theorem was given by Kono in [9], [10]. This proof naturally includes an equivariant extension of the theorem. When specialized to the case in which X is a point, the equivariant theorem reduces to the explicit Brauer induction theorem of Snaith [13], [14], [15]. More precisely, the splitting constructed by Kono coincides with that obtained algebraically by Boltje in [1] and topologically by Symonds in [17]. (See also [2]and [16].) It is the purpose of this note to clarify the relation between Segal's splitting theorem and the Brauer induction theorem in the explicit form due to Snaith, Boltje and Symonds. The methods used are from equivariant fibrewise stable homotopy theory, specifically fixed-point theory.

show abstract

Section: The Brauer-segal Theoremmentioning

confidence: 99%

Section: The Brauer-segal Theoremmentioning

confidence: 99%