What is the three-dimensional hyperbolic volume of a four-manifold? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T16:35:16Z http://mathoverflow.net/feeds/question/69819 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69819/what-is-the-three-dimensional-hyperbolic-volume-of-a-four-manifold What is the three-dimensional hyperbolic volume of a four-manifold? Bruno Martelli 2011-07-08T19:14:14Z 2012-10-24T13:59:03Z <p>Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.</p> <p>We can of course order the handles according to their index. Handles of index 0 and 1 form a connected 4-dimensional handlebody, whose boundary is a closed 3-manifold, diffeomorphic to a connected sum <code>$\#_g(S^2\times S^1)$</code> of some $g$ copies of $S^2\times S^1$ (if $g=0$ we get $S^3$). Handles of index 2 are attached to some framed link <code>$L\subset \#_g(S^2\times S^1)$</code>. Since 3- and 4-handles also form a 1-dimensional handlebody, after the attaching of the 2-handles we must necessarily obtain a 4-manifold whose boundary is again diffeomorphic to <code>$\#_h(S^2\times S^1)$</code>, for some $h$ which is not necessarily equal to $g$. The new <code>$\#_h(S^2\times S^1)$</code> is obtained by <i>surgery</i> along the framed link $L$.</p> <p>Therefore, in some sense, constructing closed orientable 4-manifolds reduces to constructing framed links <code>$L\subset \#_g(S^2\times S^1)$</code> that produce some <code>$\#_h(S^2\times S^1)$</code> via surgery. Let us define the <i> 3-dimensional hyperbolic volume </i> of a handle decomposition as the Gromov norm of $(S^2\times S^1) \setminus L$ (which is in turn the sum of the hyperbolic volumes of its pieces according to geometrization, whence the name). Let us then define the <i> 3-dimensional hyperbolic volume </i> of a closed orientable 4-manifold as the infimum of all the hyperbolic 3-dimensional volumes among all its handle decompositions. The infimum is actually a minimum because the set of 3-dimensional hyperbolic volumes is well-ordered.</p> <p>The general question is:</p> <blockquote> <p>What can we say about the 3-dimensional hyperbolic volume of a closed 4-manifold?</p> </blockquote> <p>A more specific one:</p> <blockquote> <p>Which closed 4-manifolds have zero 3-dimensional hyperbolic volume?</p> </blockquote> <p>which is equivalent to the following:</p> <blockquote> <p>Which closed 4-manifolds admit a handle decomposition such that <code>$\#_g(S^2\times S^1)\setminus L$</code> is a graph manifold?</p> </blockquote> <p>Complex projetive plane belongs to this class, and also many doubles of 2-handlebodies: if you take any link $L\subset S^2\times S^1$ whose complement is a graph manifold, you can attach 2-handles to it, and then make the double of the resulting bounded 4-manifold. The resulting double has volume zero.</p> <p>Finally, we have the following very specific question:</p> <blockquote> <p>Is there a 4-manifold with positive 3-dimensional hyperbolic volume?</p> </blockquote> <p>I would expect that most (all?) aspherical 4-manifolds have positive volume, and maybe also many simply connected ones, but I don't know the answer.</p>