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    Surgery, bordism and equivalence of 3-manifolds

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    Author
    Gerges, Amir
    Date
    1997
    Advisor
    Cochran, Tim D.
    Degree
    Doctor of Philosophy
    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.
    Keyword
    Mathematics
    Citation
    Gerges, Amir. "Surgery, bordism and equivalence of 3-manifolds." (1997) Diss., Rice University. https://hdl.handle.net/1911/19161.
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    Managed by the Digital Scholarship Services at Fondren Library, Rice University
    Physical Address: 6100 Main Street, Houston, Texas 77005
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