Surgery, bordism and equivalence of 3-manifolds

Abstract

We call two closed, oriented 3-manifolds, $M\sb0$ and $M\sb1,$ HTS-equivalent if there exists a sequence $M\sb{j\sb{i}},\ i = 1..m$ where $M\sb0 = M\sb{j\sb1}$ and $M\sb1 = M\sb{j\sb{m}}$ and $M\sb{j\sb{i}}$ is obtained from $M\sb{j\sb{i-1}}$ by performing $1/k\sb{i}$ Dehn surgery $(k\sb{i}\in\doubz)$ on a knot which is homologically trivial in $H\sb1 (M\sb{j\sb{i-1}}).$ The object of this work is to further characterize this equivalence relation. We show that saying that (i) $M\sb0$ and $M\sb1$ are HTS-equivalent, is equivalent to saying that (ii) $\exists f\sb1:M\sb1\to K(H\sb1(M\sb0),1)$ which induces an isomorphism on the first integral homology and such that if we let $f\sb0$ denote the inclusion map $M\sb0\to K(H\sb1 (M\sb0),1),$ then $f\sb{0\*}(\lbrack M\sb0\rbrack)=f\sb{1\*}(\lbrack M\sb1\rbrack)\in\Omega\sb3(K(H\sb1(M\sb0),1)).$ Where $\Omega\sb{k}(X)$ denotes the $k\sp{th}$ dimensional bordism group of X consisting of classes of k $-$ manifolds which are bordant into X. $\Omega\sb\*$ forms an extraordinary homology theory (the point axiom does not hold.) We will also show that $\Omega\sb3(X)\cong H\sb3(X).$ Moreover we show that the above conditions imply that the following holds for all n: (iii) $\exists\phi\sb1{:}H\sb1(M\sb0)\to H\sb1(M\sb1)$ an isomorphism of the first integer homology groups such that its induced isomorphisms $\phi\sbsp{n}{1}{:}H\sp1(M\sb0;\doubz\sb{n})\to H\sp1(M\sb1;\doubz\sb{n})\ (n > 0)$ satisfy the following: (a) $\langle\alpha\cup\beta\cup\gamma,\lbrack M\sb0\rbrack\rangle = \langle\phi\sbsp{n}{1}(\alpha)\cup\phi\sbsp{n}{1}(\beta)\cup\phi\sbsp{n}{1}(\gamma), \lbrack M\sb1\rbrack\rangle$ where $\phi\sbsp{n}{1}:H\sp1(M\sb0;\doubz\sb{n}) \to H\sp1(M\sb1;\doubz\sb{n})$ and $\alpha,\beta,\gamma\in H\sp1 (M\sb0;\doubz\sb{n}).$ (b) $\langle\alpha\cup B(\gamma),\lbrack M\sb0\rbrack\rangle = \langle\phi\sbsp{n}{1}(\alpha)\cup B(\phi\sbsp{n}{1}(\gamma)),$ ($M\sb1\rbrack\rangle$ where $\phi\sbsp{n}{1}\alpha,\gamma$ are as above and $B:H\sp1(\sb-;\doubz\sb{n})\to H\sp2(\sb-;\doubz\sb{n})$ is the Bockstein operator associated with the short exact sequence: $0\to\doubz\sb{n}{\buildrel {n}\over{\to}}\doubz\sb{n\sp2}\to\doubz\sb{n}\to 0$ Then we show that satisfying (iii) for a subset of the integers (namely when n is a power of a prime) implies (ii). As corollaries of this theorem, we get for instance that all 3-manifolds with first homology group isomorphic to $\doubz$ or to $\doubz\oplus\doubz$ are HTS-equivalent. This is no longer true for rank higher than 2. We also show that any two lens spaces are HTS-equivalent if and only if they are homotopy equivalent via an orientation preserving map. Finally we mention some generalizations and future directions.