Groups of Automorphisms of Manifolds

Burghelea, Dan; Lashof, R.; Rothenberg, Melvin

doi:10.1007/bfb0079981

Cited by 60 publications

(56 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Its proof involves a mixture of homotopy theory, surgery, and Morlet's disjunction lemma (Morlet [32], Burghelea-Lashof-Rothenburg [6]). (A square is j -cartesian if the map from its initial term to the homotopy pullback of the other terms is j -connected.…”

Section: Disjunctionmentioning

confidence: 99%

Homotopical intersection theory I

Klein

Williams

2007

Geom. Topol.

View full text Add to dashboard Cite

We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn [17], but our proofs are fundamentally different. Errata Minor errors were corrected on page 967 (18 February 2008).

show abstract

Section: Disjunctionmentioning

confidence: 99%

Homotopical intersection theory I

Klein

Williams

2007

Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…T is the semidirect product of subgroups H and T, where U is a Bieberbach group of rank n -1; or (2) r = B*d C; i.e. is the amalgamated free product of subgroups of rank n-1, where D has index 2 in both B and C; or (3) there is an infinite sequence of positive integers Sn with Sn = lmod|G| (|G| = order of G) such that any hyperelementary subgroup ofFs which projects onto G via the canonical map T -> G projects isomorphically to G.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…If £ is a hyperelementary subgroup of Ts which projects via p to a proper subgroup of G, then the holonomy group of p~1(E) has order less than the order of G; thus, Whj(p*1(S))(p) = 0 by the induction hypothesis. Otherwise E projects isomorphically to G by Theorem 2.8 (3). Since all such subgroups are conjugate [5] it suffices to consider one of them.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

On the Higher Whitehead Groups of a Bieberbach Group

Nicas¹

1985

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

ABSTRACT. Let T be a Bieberbach group, i.e. the fundamental group of a compact flat Riemannian manifold. In this paper we show that if p > 2 is a prime, then the p-torsion subgroup of Wh¿(r) vanishes for 0 < i < 2p -2, where Wh¿(r) is the ¿th higher Whitehead group of T. The proof involves Farrell and Hsiang's structure theorem for Bieberbach groups, parametrized surgery, pseudoisotopy, and Waldhausen's algebraic if-theory of spaces. Introduction.A closed aspherical manifold is a closed manifold whose universal covering space is contractible. There is the following conjecture concerning the algebraic if-theory of such manifolds.Conjecture. Let T be the fundamental group of a closed aspherical manifold.Then Wh¿(r) = 0 for i > 0, where Wh¿ is the ¿th higher Whitehead group of T. The notation A(pj is used here to denote the p-torsion subgroup of the abelian group A. The main technical ingredient is Proposition 1.5 which asserts a certain vanishing property for the transfer map of 7Tj(Wh (M)) for compact flat Riemannian M. Using a different approach, Frank Quinn has announced that he has developed new techniques which will prove Wh¿(r) = 0, i > 0, for a Bieberbach group T.

show abstract

“…THEOREM 1.2. (MorletV'Lemme de Disjunction", [1,7]). LetD Letting N = S n~1 xI, so that fe = n-2, Theorem 1.1 shows immediately that Theorem 1.3 is best possible.…”

mentioning

confidence: 99%

“…Letting D 1 = * x IC S"" 1 x I = N, the fact (proved via an Alexander isotopy argument, cf. [1,5]) that…”

mentioning

confidence: 99%