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Controlled homotopy equivalences and structure sets of manifolds


Authors: Friedrich Hegenbarth and Dušan Repovš
Journal: Proc. Amer. Math. Soc. 142 (2014), 3987-3999
MSC (2010): Primary 57R67, 57P10, 57R65; Secondary 55N20, 55M05
DOI: https://doi.org/10.1090/S0002-9939-2014-12131-9
Published electronically: July 16, 2014
MathSciNet review: 3251739
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Abstract: For a closed topological $ n$-manifold $ K$ and a map $ p:K\to B$ inducing an isomorphism $ \pi _{1}(K)\to \pi _{1}(B)$, there is a canonically defined morphism $ b:H_{n+1}(B,K,\mathbb{L} )\to \mathcal {S}(K)$, where $ \mathbb{L} $ is the periodic simply connected surgery spectrum and $ \mathcal {S}(K)$ is the topological structure set. We construct a refinement $ a:H_{n+1}^{+}(B,K,\mathbb{L} )\to \mathcal {S}_{\varepsilon ,\delta }(K)$ in the case when $ p$ is $ UV^{1}$, and we show that $ a$ is bijective if $ B$ is a finite-dimensional compact metric ANR. Here, $ H_{n+1}^{+}(B,K,\mathbb{L} )\subset H_{n+1}(B,K,\mathbb{L} )$, and $ \mathcal {S}_{\varepsilon ,\delta }(K)$ is the controlled structure set. We show that the Pedersen-Quinn-Ranicki controlled surgery sequence is equivalent to the exact $ \mathbb{L} $-homology sequence of the map $ p:K \to B$, i.e. that

$\displaystyle H_{n+1}(B,\mathbb{L} )\to H_{n+1}^{+}(B,K,\mathbb{L} )\to H_{n}(K,\mathbb{L} ^{+})\to H_{n}(B,\mathbb{L} ), \ \mathbb{L} ^{+}\to \mathbb{L},$    

is the connected covering spectrum of $ \mathbb{L} $. By taking for $ B$ various stages of the Postnikov tower of $ K$, one obtains an interesting filtration of the controlled structure set.

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Additional Information

Friedrich Hegenbarth
Affiliation: Department of Mathematics, University of Milano, Via C. Saldini 50, 02130 Milano, Italy
Email: friedrich.hegenbarth@mat.unimi.it

Dušan Repovš
Affiliation: Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia
Email: dusan.repovs@guest.arnes.si

DOI: https://doi.org/10.1090/S0002-9939-2014-12131-9
Keywords: Controlled surgery, $UV^{1}$--property, $\mathbb{L} $--homotopy, $\mathbb{L} $--homology, controlled structure set, Wall obstruction
Received by editor(s): October 28, 2010
Received by editor(s) in revised form: December 3, 2012
Published electronically: July 16, 2014
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2014 American Mathematical Society

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