Controlled homotopy equivalences and structure sets of manifolds
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- by Friedrich Hegenbarth and Dušan Repovš PDF
- Proc. Amer. Math. Soc. 142 (2014), 3987-3999 Request permission
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 \begin{equation*}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}, \end{equation*} 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.References
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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
- MR Author ID: 147135
- ORCID: 0000-0002-6643-1271
- Email: dusan.repovs@guest.arnes.si
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3251739