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Hegenbarth, Friedrich; Repovš, Dušan
Mediterranean journal of mathematics, 06/2019, Volume: 16, Issue: 3Journal Article
This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map ( f , b ) : M n → X n with control map q : X n → B to complete controlled surgery is an element σ c ( f , b ) ∈ H n ( B , L ) , where M n , X n are topological manifolds of dimension n ≥ 5 . Our proof uses essentially the geometrically defined L -spectrum as described by Nicas (going back to Quinn) and some well-known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describe the assembly map H n ( B , L ) → L n ( π 1 ( B ) ) in terms of forms in the case n ≡ 0 ( 4 ) . Finally, we explicitly determine the canonical map H n ( B , L ) → H n ( B , L 0 ) .
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