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13
Automorphisms of manifolds and algebraic Ktheory: I, KTheory 1
 Stanford University, Stanford
, 1988
"... Abstract. Let M be a closed topological n–manifold, and let S(M) be the moduli space of closed topological manifolds equipped with a homotopy equivalence to M. We give an algebraic description of S(M) in the hcobordism stable range, assuming n ≥ 5. (That is, we produce a highly connected map from S ..."
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Cited by 23 (4 self)
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Abstract. Let M be a closed topological n–manifold, and let S(M) be the moduli space of closed topological manifolds equipped with a homotopy equivalence to M. We give an algebraic description of S(M) in the hcobordism stable range, assuming n ≥ 5. (That is, we produce a highly connected map from S(M) to another space having an algebraic description.) The algebraic description is in terms of L–theory, Waldhausen’s algebraic K–theory of spaces, and a natural transformation Ξ (constructed in our paper [WW2]) from L–theory to the Tate cohomology of Z2 acting on K–theory. We develop a parallel theory for the moduli space S(τ) of Rn –bundles on M equipped with an ”equivalence ” to the tangent bundle τ of M. (The equivalence is a stable fiber homotopy equivalence of the corresponding spherical fibrations.) Results about moduli spaces of smooth manifolds can be obtained by combining the calculations of S(M) and S(τ). We have attempted to make this paper as self–contained as possible by summarizing results from the earlier papers in the series where necessary.
On The Calculation Of Unil
 Adv. Math. 195 (2005) 205–258 (2005), eprint http://arXiv.org/abs/math.AT/0304016
, 2003
"... Cappell's codimension 1 splitting obstruction surgery group UNiln is a direct summand of the Wall surgery obstruction group of an amalgamated free product. For any ring with involution R we use the quadratic Poincare cobordism formulation of the Lgroups to prove that Ln(R[x]) = Ln (R) #UNiln ..."
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Cited by 16 (2 self)
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Cappell's codimension 1 splitting obstruction surgery group UNiln is a direct summand of the Wall surgery obstruction group of an amalgamated free product. For any ring with involution R we use the quadratic Poincare cobordism formulation of the Lgroups to prove that Ln(R[x]) = Ln (R) #UNiln (R; R, R) .
Generalized Arf invariants in algebraic Ltheory
"... Abstract. The difference between the quadratic Lgroups L∗(A) and the symmetric Lgroups L ∗ (A) of a ring with involution A is detected by generalized Arf invariants. The special case A = Z[x] gives a complete set of invariants for the Cappell UNilgroups UNil∗(Z; Z, Z) for the infinite dihedral gr ..."
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Cited by 9 (0 self)
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Abstract. The difference between the quadratic Lgroups L∗(A) and the symmetric Lgroups L ∗ (A) of a ring with involution A is detected by generalized Arf invariants. The special case A = Z[x] gives a complete set of invariants for the Cappell UNilgroups UNil∗(Z; Z, Z) for the infinite dihedral group D ∞ = Z2 ∗Z2, extending the results of Connolly and Ranicki [10], Connolly and Davis [8].
Poincaré Sheaves on Topological Spaces
 Trans. Amer. Math. Soc
"... We define the notion of a Poincar'e complex of sheaves over a topological space. Global surgery invariants of spaces and maps are expressed as the assembly of locally defined surgery invariants. For simplicial complexes we recover the theory of Poincar'e cycles of Ranicki. For more general ..."
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Cited by 5 (1 self)
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We define the notion of a Poincar'e complex of sheaves over a topological space. Global surgery invariants of spaces and maps are expressed as the assembly of locally defined surgery invariants. For simplicial complexes we recover the theory of Poincar'e cycles of Ranicki. For more general spaces we obtain local surgery invariants in the absence of triangulations. 1.
ALGEBRAIC POINCARÉ COBORDISM
, 2000
"... The object of this paper is to give a reasonably leisurely account of the algebraic Poincaré cobordism theory of Ranicki [15],[16] and the further development due to ..."
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Cited by 1 (0 self)
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The object of this paper is to give a reasonably leisurely account of the algebraic Poincaré cobordism theory of Ranicki [15],[16] and the further development due to
www.elsevier.com/locate/aim GeneralizedArf invariants in algebraic Ltheory
, 2005
"... The difference between the quadratic Lgroups L∗(A) and the symmetric Lgroups L∗(A) of a ring with involution A is detected by generalized Arf invariants. The special case A = Z[x] gives a complete set of invariants for the Cappell UNilgroups UNil∗(Z;Z,Z) for the infinite dihedral group D ∞ = Z2 ..."
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The difference between the quadratic Lgroups L∗(A) and the symmetric Lgroups L∗(A) of a ring with involution A is detected by generalized Arf invariants. The special case A = Z[x] gives a complete set of invariants for the Cappell UNilgroups UNil∗(Z;Z,Z) for the infinite dihedral group D ∞ = Z2 ∗ Z2, extending the results of Connolly and Ranicki [Adv.
The Kervaire invariant and surgery theory
"... Abstract. We give an expository account of the development of the Kervaire invariant and its generalizations with emphasis on its applications to surgery and, in particular, to the existence of stably parallelizable manifolds with Kervaire invariant one. 1. ..."
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Abstract. We give an expository account of the development of the Kervaire invariant and its generalizations with emphasis on its applications to surgery and, in particular, to the existence of stably parallelizable manifolds with Kervaire invariant one. 1.
www.elsevier.com/locate/aim Generalized Arf invariants in algebraic Ltheory
, 2005
"... The difference between the quadratic Lgroups L∗(A) and the symmetric Lgroups L ∗ (A) of a ring with involution A is detected by generalized Arf invariants. The special case A = Z[x] gives a complete set of invariants for the Cappell UNilgroups UNil∗(Z; Z, Z) for the infinite dihedral group D ∞ = ..."
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The difference between the quadratic Lgroups L∗(A) and the symmetric Lgroups L ∗ (A) of a ring with involution A is detected by generalized Arf invariants. The special case A = Z[x] gives a complete set of invariants for the Cappell UNilgroups UNil∗(Z; Z, Z) for the infinite dihedral group D ∞ = Z2 ∗ Z2, extending the results of Connolly and Ranicki [Adv.