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We show that we can maintain up to polylogarithmic edge connectivity for a fully-dynamic graph in $$\ifmmode\expandafter\tilde\else\expandafter\~\fi{O}{\left( {{\sqrt n }} \right)}$$ worst-case time per edge insertion or deletion. Within logarithmic factors, this matches the best time bound for 1-edge connectivity. Previously, no o(n) bound was known for edge connectivity above 3, and even for 3-edge connectivity, the best update time was O(n2/3), dating back to FOCS'92. Our algorithm maintains a concrete min-cut in terms of a pointer to a tree spanning one side of the cut plus ability to list the cut edges in O(log n) time per edge. By dealing with polylogarithmic edge connectivity, we immediately get a sampling based expected factor (1+o(1)) approximation to general edge connectivity in $$\ifmmode\expandafter\tilde\else\expandafter\~\fi{O}{\left( {{\sqrt n }} \right)}$$ time per edge insertion or deletion. This algorithm also maintains a pointer to one side of a near-minimal cut, but if we want to list the cut edges in O(log n) time per edge, the update time increases to $$\ifmmode\expandafter\tilde\else\expandafter\~\fi{O}{\left( {{\sqrt m }} \right)}$$.