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Genus g surface
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<p>In mathematics, a genus <i>g</i> surface (also known as a <i>g</i>-torus or <i>g</i>-holed torus) is a <a href="/facts/Surface_(topology)/jK1b1KIG">surface</a> formed by the <a href="/facts/Connected_sum/TycjIkQe">connected sum</a> of <i>g</i> distinct <a href="/facts/Torus/J9SkaV1W">tori</a>: the interior of a disk is removed from each of <i>g</i> distinct tori and the boundaries of the <i>g</i> many disks are identified (glued together), forming a <i>g</i>-torus. The <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> of such a surface is <i>g</i>. </p><p>A genus <i>g</i> surface is a <a href="/facts/Two-dimensional/f4b397W8">two-dimensional</a> <a href="/facts/Manifold/rXPERJtu">manifold</a>. The <a href="/facts/Surface_(topology)/jK1b1KIG">classification theorem for surfaces</a> states that every <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Connected_space/5CUTxfoA">connected</a> two-dimensional manifold is <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphic</a> to either the sphere, the connected sum of tori, or the connected sum of <a href="/facts/Real_projective_plane/gGVd0ctY">real projective planes</a>. </p>