A　bisection　of　a　graph　is　a　bipartition　of　its　vertex　set　in　which　the　number　of　vertices　in　the　two　parts　differ　by　at　most　1,　and　its　size　is　the　number　of　edges　which　go　across　the　two　parts.　In　this　paper,　motivated　by　several　known　results　of　Alon,　Bollobas,　Krivelevich　and　Sudakov　about　Max-Cut,　we　study　maximum　bisections　of　graphs　without　short　even　cycles.　Let　G　be　a　graph　on　medges　without　cycles　of　length　4　and　6.　We　first　extend　a　well-known　result　of　Shearer　on　maximum　cuts　to　bisections　and　show　that　if　G　has　a　perfect　matching　and　degree　sequence　d(1),　...,　d(n),　then　G　admits　a　bisection　of　size　at　least　m/2　+　Omega　(Sigma(n)(i=1)　root　d(i)).　This　is　tight　for　certain　polarity　graphs.　Together　with　a　technique　of　Nikiforov,　we　prove　that　if　G　also　contains　no　cycle　of　length　2k　＞=　6　then　G　either　has　a　large　bisection　or　is　nearly　bipartite.　As　a　corollary,　if　G　has　a　matching　of　size　circle　minus(n),　then　G　admits　a　bisection　of　size　at　least　m/2　+　Omega　(m((2k+1)/(2k+2))　and　that　this　is　tight　for　2k　is　an　element　of　{6,　10};　if　G　has　a　matching　of　size　o(n),　then　the　bound　remains　valid　for　Gwith　minimum　degree　at　least　2.　(C)　2021　Elsevier　Inc.　All　rights　reserved.