{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T12:25:55Z","timestamp":1765369555285,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In 1970, Walkup completely described the set of $f$-vectors for the four $3$-manifolds $S^3$, $S^2\\rlap{\\times}\\_\\;S^1$, $S^2\\!\\times\\!S^1$, and ${\\Bbb R}{\\bf P}^{\\,3}$. We improve one of Walkup's main restricting inequalities on the set of $f$-vectors of $3$-manifolds. As a consequence of a bound by Novik and Swartz, we also derive a new lower bound on the number of vertices that are needed for a combinatorial $d$-manifold in terms of its $\\beta_1$-coefficient, which partially settles a conjecture of K\u00fchnel.  Enumerative results and a search for small triangulations with bistellar flips allow us, in combination with the new bounds, to completely determine the set of $f$-vectors for twenty further $3$-manifolds, that is, for the connected sums of sphere bundles $(S^2\\!\\times\\!S^1)^{\\# k}$ and twisted sphere bundles $(S^2\\rlap{\\times}\\_\\;S^1)^{\\# k}$, where $k=2,3,4,5,6,7,8,10,11,14$. For many more $3$-manifolds of different geometric types we provide small triangulations and a partial description of their set of $f$-vectors.  Moreover, we show that the $3$-manifold ${\\Bbb R}{\\bf P}^{\\,3}\\#\\,{\\Bbb R}{\\bf P}^{\\,3}$ has (at least) two different minimal $g$-vectors.<\/jats:p>","DOI":"10.37236\/79","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:22:01Z","timestamp":1578716521000},"source":"Crossref","is-referenced-by-count":19,"title":["$f$-Vectors of $3$-Manifolds"],"prefix":"10.37236","volume":"16","author":[{"given":"Frank H.","family":"Lutz","sequence":"first","affiliation":[]},{"given":"Thom","family":"Sulanke","sequence":"additional","affiliation":[]},{"given":"Ed","family":"Swartz","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,5,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i2r13\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i2r13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:53:29Z","timestamp":1579316009000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i2r13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,5,22]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2009,2,11]]}},"URL":"https:\/\/doi.org\/10.37236\/79","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,5,22]]},"article-number":"R13"}}