Middle Dimension Research Articles

Alexander Modules

The Alexander modules of a link are the homology groups of the universal abelian cover of the complement of the link. For a link of $n$-spheres in ${S^{n + 2}}$, we show that, if $n \geqslant 2$, the Alexander modules ${A_2}, \ldots ,{A_n}$ and the torsion submodule of ${A_1}$ are all of type $L$. This leads to a characterization, below the middle dimension, of the polynomial invariants of the link. These results were previously proven for the special case of boundary links.

Alexander modules

The Alexander modules of a link are the homology groups of the universal abelian cover of the complement of the link. For a link of n n -spheres in S n + 2 {S^{n + 2}} , we show that, if n ⩾ 2 n \geqslant 2 , the Alexander modules A 2 , … , A n {A_2}, \ldots ,{A_n} and the torsion submodule of A 1 {A_1} are all of type L L . This leads to a characterization, below the middle dimension, of the polynomial invariants of the link. These results were previously proven for the special case of boundary links.

L 2-index and the Selberg trace formula

A method is developed for computing the L 2-index of a “locally symmetric” elliptic differential operator D Γ , acting on a locally symmetric manifold M Γ = ΓβG K with G semisimple of real-rank one and Γ of finite co-volume, based on applying the Selberg trace formula to the difference of the two heat kernels associated to D Γ . The applications include an extension of the Osborne-Warner multiplicity formula to certain non-integrable discrete series—derived from the L 2-spinor formula, and showing the existence, in some cases, of non-invariant L 2-cohomology classes in the middle dimension—via the L 2-signature formula.

Middle dimension homologies of a codimension 2 submanifold. II

This paper is devoted to the estimates for the middle dimension homologies of codimension 2 submanifoldA of a smooth manifold X. Proofs are based on the information about homologies of the ramified covering of X branched alongA. This information is extracted from the Atiyah-Singer signature formulas and from some corollaries of the Smith theory.

Homology circles and knot complements

BY A lzonzology circle we mean a CW complex C (or a Kan complex) for which there is an isomorphism H,(C. 2) = H,(S’, 2) where S’ denotes the usual circle. Such spaces are always equipped with a map C + S’ which induces that isomorphism. The best known examples of homology circles are knot complements C = S”+‘N(K) where N(K) is an open tubular neighborhood of a submanifold K homeomorphic to the n-sphere S”. Here we are interested only in the homotop_v type of homology circles. Indeed it was proven by Browder, Lashof and Shaneson [9] that given the homotopy type of the pair (C, aC) it corresponds to, at most, two isotopy types of knots-if a,C = Z. From this point of view the homotopy type of (compact) homology circles is of major importance. Kervaire has done some fundamental work on the first non-trivial homotopy group ai(S”, C)[8] and we shall be interested in the higher homotopy groups. Kervaire also notes that “almost” arbitrary compact homology circles can be realized as knot complements, up to the middle dimension [see 5.12 below]. More generally, the present work is a continuation of the general work on homology isomorphism: How can one analyze and construct “all” homology circles? What are the possible homotopy groups and “k-invariants” of such spaces? What precise part of the (n + I)-type of C is determined by the n-type and what can be freely chosen? Such questions have relatively simple answers when one deals with e.g. acyclic spaces [3]. It is surprising to see that the case of homology circles turns out to be of the same general nature. But the details are more difficult since the fundamental group is not perfect. In the present paper we only deal with two types of homology circles: the (homotopy) finite case and the nilpotent case. We give a “classification” of these. It turns out that most of the important features of the general case arise already here and they assume here a concise and simple form. Roughly speaking the homotopy groups 7r” = TX appear naturally as extensions 0 + In, --, 7~, * nn IIT, + 0 where TV” is “a-perfect” and can be almost arbitrarily chosen and rr,,/I7r, depends on the (homologies of the) choices made in the lower dimension: In, for j <n. These considerations lead us to a general theorem which determines the structure of the second non-trivial homotopy group of a knot complement, in terms of the first non-trivial one (see 5.2). A word about the category in which we work: By “space” and “map” we mean objects and morphisms in the pointed category of CW complexes or Kan complexes. This small abuse of notation is harmless because we are interested in questions of homotopy type and the-geometric realizations provides an equivalence between the two categories [lo]. The paper is organized as follows. After formulating below the main results in general terms we recall some basic concepts and their properties in § I. $2 is devoted to the analysis of nilpotent homology circles. In 03 and 04 we explain and prove theorems relating to (homotopy) finite homology circles. Some applications and geometric remarks are given in $5.

Bordism Invariants of Intersections of Submanifolds

This paper characterizes certain geometric intersection problems in terms of bordism obstructions. These obstructions give a setting in which to study such things as parametrized $h$-cobordisms (pseudoisotopy), and surgery above the middle dimension and on fibrations, where such intersection problems arise.

Bordism invariants of intersections of submanifolds

This paper characterizes certain geometric intersection problems in terms of bordism obstructions. These obstructions give a setting in which to study such things as parametrized $h$-cobordisms (pseudoisotopy), and surgery above the middle dimension and on fibrations, where such intersection problems arise.

Groups of Embedded Manifolds

This paper defines a O(Mn, v0) which generalizes the of framed homotopy n-spheres in Sn+k. Let Mn be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let v, be the normal bundle of some imbedding p: Mn -_ Sn + k, where 2k _ n + 3. Then O(Mn, v(,) is the set of h-cobordism classes of triples (F, Vn, f), where F: Sn + k T(v0) is a map which is transverse regular on M, Vn=F-1(Mn), and f=Ff Vn is a homotopy equivalence. (T(v,) is the Thom complex of v,.) There is a natural structure on 0(Mn, v1,), and O(Mn, v1,) fits into an exact sequence similar to that for the framed homotopy n-spheres. This paper attempts to generalize in a natural way a well-known exact sequence concerning framed homotopy spheres which is contained in the work of Novikov [11], Kervaire-Milnor [7], and Levine [10]. The author stumbled onto these results partly because of his efforts to prove imbedding theorems for manifolds in the metastable range, and partly because of his recent work on Browder-Novikov theory for maps of degree d, Idl I=0 (see [2]). ?2 describes the basic constructions used in this paper. The group of embedded manifolds, 6(M, v,), is defined in ?3. A fairly simple description of that is given towards the end of that section. ?3 also contains the main results about O(M, v,). In ?4 we discuss a few interesting open problems. The author would like to thank the referee for some helpful suggestions. 1. Notation. All manifolds will be C O, compact, and oriented. Maps will be transverse to boundaries. If Mn is a connected closed manifold, let [M] E HnM denote the orientation class. Recall that f: Vn Mn is said to have degree d, i.e., degf=d, if f*([V]) =d[M], where f*: HnV HnM is the map induced byf on the integral homology groups. As usual, Dk denotes the closed unit ball in Euclidean k-space Rk, i.e., Dk={(yl, Yy) Rk|y2+I*...*+y2 O} and D_ ={(Y1, ,Yk+) eRk+l y2+ *?. +y2+ 1?=1,Y?<0}. We have natural inclusions S?kSk+l and DkCDk+l. Let e=(l 0) E SO?Sk. Received by the editors September 5, 1969. AMS 1970 subject classifications. Primary 57D55, 57D99; Secondary 57D60.

Groups of embedded manifolds

This paper defines a group O(Mn, v0) which generalizes the group of framed homotopy n-spheres in Sn+k. Let Mn be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let v, be the normal bundle of some imbedding p: Mn -_ Sn + k, where 2k _ n + 3. Then O(Mn, v(,) is the set of h-cobordism classes of triples (F, Vn, f), where F: Sn + k T(v0) is a map which is transverse regular on M, Vn=F-1(Mn), and f=Ff Vn is a homotopy equivalence. (T(v,) is the Thom complex of v,.) There is a natural group structure on 0(Mn, v1,), and O(Mn, v1,) fits into an exact sequence similar to that for the framed homotopy n-spheres. This paper attempts to generalize in a natural way a well-known exact sequence concerning framed homotopy spheres which is contained in the work of Novikov [11], Kervaire-Milnor [7], and Levine [10]. The author stumbled onto these results partly because of his efforts to prove imbedding theorems for manifolds in the metastable range, and partly because of his recent work on Browder-Novikov theory for maps of degree d, Idl I=0 (see [2]). ?2 describes the basic constructions used in this paper. The "group of embedded manifolds", 6(M, v,), is defined in ?3. A fairly simple description of that group is given towards the end of that section. ?3 also contains the main results about O(M, v,). In ?4 we discuss a few interesting open problems. The author would like to thank the referee for some helpful suggestions. 1. Notation. All manifolds will be C O, compact, and oriented. Maps will be transverse to boundaries. If Mn is a connected closed manifold, let [M] E HnM denote the orientation class. Recall that f: Vn Mn is said to have degree d, i.e., degf=d, if f*([V]) =d[M], where f*: HnV HnM is the map induced byf on the integral homology groups. As usual, Dk denotes the closed unit ball in Euclidean k-space Rk, i.e., Dk={(yl, Yy) Rk|y2+I*...*+y2 O} and D_ ={(Y1, ,Yk+) eRk+l y2+ *?. +y2+ 1?=1,Y?<0}. We have natural inclusions S?kSk+l and DkCDk+l. Let e=(l 0) E SO?Sk. Received by the editors September 5, 1969. AMS 1970 subject classifications. Primary 57D55, 57D99; Secondary 57D60.