On the Surgery Theory for Filtered Manifolds

Alberto Cavicchioli,Yurij V Muranov,Friedrich Hegenbarth,Fulvia Spaggiari

doi:10.13189/ms.2013.010405

Alberto Cavicchioli, Yurij V Muranov + Show 2 more

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https://doi.org/10.13189/ms.2013.010405

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Abstract

In this paper we describe some relations between various structure sets which arise naturally for a Browder-Livesay ltration of a closed topological mani- fold. We use the algebraic surgery theory of Ranicki for realizing the surgery groups and natural maps on the spectrum level. We obtain also new relations between Browder{Quinn surgery obstruction groups and structure sets. Finally we illustrate several examples and applications.

Highlights

Denote by Fi (0 ≤ i ≤ k − 1) the square of fundamental groups in the splitting problem for the manifold pair (Xi, Xi+1) fitting into filtration (1.8)
Let Xn be a closed connected n-dimensional topological manifold with fundamental group π = π1(X)
In this paper we describe some relations between various structure sets which arise naturally for a Browder-Livesay filtration of a closed topological manifold

Summary

Introduction

Denote by Fi (0 ≤ i ≤ k − 1) the square of fundamental groups in the splitting problem for the manifold pair (Xi, Xi+1) fitting into filtration (1.8). For a Browder-Livesay filtration X (1.8) and for a simple homotopy equivalence f : M → X there is an obstruction in the group LSFn−k(X ) to find an striangulation of X in the homotopy class of the map f [3]. In this paper we describe new relations between structure sets and various obstruction groups which arise for a Browder-Livesay filtration.

Results

Conclusion