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The smooth Whitehead spectrum of a point at odd regular primes. (English) Zbl 1130.19300

Summary: Let \(p\) be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for \(K(\mathbb Z[1/p])\) at \(p\). Then the \(p\)-primary homotopy type of the smooth Whitehead spectrum \(\mathrm{Wh}(*)\) is described. A suspended copy of the cokernel-of-\(J\) spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted \(S^1\)-transfer map \(t: \Sigma \mathrm{CP}^\infty\to S\). The homotopy groups of \(\mathrm{Wh}(*)\) are determined in a range of degrees, and the cohomology of \(\mathrm{Wh}(*)\) is expressed as an \(A\)-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.

MSC:

19D10 Algebraic \(K\)-theory of spaces
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
55P42 Stable homotopy theory, spectra
55Q52 Homotopy groups of special spaces
57R80 \(h\)- and \(s\)-cobordism
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