Finding a boundary for a Hilbert cube manifold

We continue our study of ends non-compact manifolds. The over-arching aim is to provide an appropriate generalization of Siebenmann's famous collaring theorem that applies to manifolds having non-stable fundamental group systems at infinity. In this paper a primary goal is finally achieved; namely, a complete characterization of pseudo-collarability for manifolds of dimension at least 6. 57N15, 57Q12; 57R65, 57Q10

show abstract

“…W i is a homotopy equivalence. For any such cobordism, 1 .B i / ! 1 .W i / is surjective and has perfect kernel (again see [8, Theorem 2.5]).…”

Section: Topology Of Ends Of Manifoldsmentioning

confidence: 99%

“…Thus we may apply Theorem 3.2 to .R i ; @U i ; @U iC1 / to obtain a plus cobordism .W i ; A i ; @U iC1 / embedded in R i which is the identity on @U iC1 and for which ker . 1 …”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Manifolds with non-stable fundamental groups at infinity, III

Guilbault

Tinsley

2006

Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…When n ≥ 6 and π 1 (ε (M n )) is semistable, we will see σ ∞ (M n ) arise naturally-without reference to the Wall finiteness obstruction-as an obstruction to pseudo-collarability (see Section 8). For a more general treatment of this obstruction-which, among other things, shows that K 0 (π 1 (ε (M n ))) and σ ∞ (M n ) are independent of the choice of {U i }-we refer the reader to [6].…”

Section: Ends Of Manifolds: Definitions and Background Informationmentioning

confidence: 99%

Manifolds with non-stable fundamental groups at infinity

Guilbault

Tinsley

2000

Geom. Topol.

View full text Add to dashboard Cite

The notion of an open collar is generalized to that of a pseudo-collar. Important properties and examples are discussed. The main result gives conditions which guarantee the existence of a pseudo-collar structure on the end of an open n-manifold (n ≥ 7). This paper may be viewed as a generalization of Siebenmann's famous collaring theorem to open manifolds with non-stable fundamental group systems at infinity. AMS Classification numbers Primary: 57N15, 57Q12Secondary: 57R65, 57Q10

show abstract

“…To this end, fix a proper ü7F"-surjection g : M -> X onto a ß-manifold X. Then choose a compact cover {Xi}i€(û of X consisting of ß-manifold with X¡ c intX,+1 such that FrZ, is a Z-set in both X, and X \ inXXj, i e co (see [C2,CS]). For each i e co, by the relative triangulation theorem for ß-manifolds [C3], we may assume that X = P x Q, Xj = P[ x Q, and X \ intX, = P2lx Q for a locally finite polyhedron P and closed subpolyhedra P{, P[ c P. Note that P[ n P¡ is a Z-set in P¡ .…”

Section: Characterization Of íZ^"1-manifoldsmentioning

confidence: 99%