dc.description.abstract 
We call two closed, oriented 3manifolds, $M\sb0$ and $M\sb1,$ HTSequivalent 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 $M\sb{j\sb{i1}}$ by performing $1/k\sb{i}$ Dehn surgery $(k\sb{i}\in\doubz)$ on a knot which is homologically trivial in $H\sb1 (M\sb{j\sb{i1}}).$ The object of this work is to further characterize this equivalence relation. We show that saying that (i) $M\sb0$ and $M\sb1$ are HTSequivalent, 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 3manifolds with first homology group isomorphic to $\doubz$ or to $\doubz\oplus\doubz$ are HTSequivalent. This is no longer true for rank higher than 2. We also show that any two lens spaces are HTSequivalent if and only if they are homotopy equivalent via an orientation preserving map.
Finally we mention some generalizations and future directions.
