On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds

Farrell, F. T.; Hsiang, W. C.

doi:10.1090/pspum/032.1/520509

Cited by 77 publications

(90 citation statements)

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“…(/c)-trivial subspaces are the same. Then the following vanishing theorem applies: If SG" is the algebraic group whose Q-points are SLn(k) and whose Z-points are SL"(C) and p is a nontrivial irreducible algebraic representation of SG"(R), then Hp(SLn(0); p) = 0 forp in a stable range 0 < p < cp(«) where <p(«) tends to infinity with n [1,3]. Then Now b and p induce homology isomorphisms between total spaces and, by the comparison theorem for spectral sequences [13], isomorphisms of the spectral sequences.…”

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confidence: 99%

Rational algebraic 𝐾-theory of certain truncated polynomial rings

Staffeldt¹

1985

Proc. Amer. Math. Soc.

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Abstract.In this paper we derive a formula for rationalized algebraic A'-theory of certain overlings of rings of integers in number fields. Truncated polynomial algebras are examples. Our method is homological calculation which is facilitated by some basic rational homotopy theory and interpreted in terms of the cyclic homology theory of algebras invented by Alain Connes.The object of this paper is to compute, in terms of A. Connes' cyclic homology and the rational algebraic /^-theory of a ring of integers 0 in a number field the dual numbers over 0. Using a classical Lie algebra homology calculation based on invariant theory, but no cyclic homology, he found that for the dual numbers dimg^ = d, or = 0, if q is odd, or even. In §2 of this paper we sketch a computation in cyclic homology relevant to the case A = 0[T]/(T" + 1) following the steps of a computation of the rational algebraic ^-theory of the space CP" shown to us by T. Goodwillie. It turns out that dim V = n ■ d, or =0, depending on whether or not q is odd, or even.The proof of this theorem, together with the proof of the theorem of Loday and Quillen relating Lie algebra homology and cyclic homology, actually gives when Z = 0 a chain of natural isomorphisms linking the rationalized relative A"-group Kq+X(A -^Z) = 7r9(fibre(BGL + (/l) -» BGL(Z)+)) ® Q

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confidence: 99%

Rational algebraic 𝐾-theory of certain truncated polynomial rings

Staffeldt¹

1985

Proc. Amer. Math. Soc.

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“…Waldhausen proved in [51] that when X = M is a compact smooth manifold there is a homotopy equivalence For example, Farrell and Hsiang [14] show that π m C(D n ) ⊗ Q has rank 1 in all degrees m ≡ 3 mod 4, and rank 0 in other degrees, for n sufficiently large with respect to m. From this they deduce that π m DIF F (D n ) ⊗ Q has rank 1 for m ≡ 3 mod 4 and n odd, and rank 0 otherwise, always assuming that m is in the concordance stable range for D n .…”

Section: The Stable Parametrized H-cobordism Theoremmentioning

confidence: 99%

The smooth Whitehead spectrum of a point at odd regular primes

Rognes

2003

Geom. Topol.

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Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K (Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum W h( * ) 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 : ΣCP ∞ → S . The homotopy groups of W h( * ) are determined in a range of degrees, and the cohomology of W h( * ) 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.

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“…Such theorems for twisted Laurent rings were later provided by Farrell and Hsiang ([FarHs2], [FarHs3]) for algebraic K-theory, by Cappell [Cap2] and Ranicki [Ran3] for algebraic L-theory, and by Pimsner-Voiculescu [PimV] for the K-theory of C * -algebras. Specific applications to the Novikov Conjecture were provided in [FarHs4], [FarHs5] and in [Ros2], [Ros4]. 8 Novikov has pointed out to us that he was referring here to (relative) invariants of degree-one normal maps, which are easier to define than (absolute) invariants for closed manifolds.…”

Section: The Original Statement Of the Novikov Conjecturementioning

confidence: 99%