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Underdoped cuprates exhibit a normal-state pseudogap, and their spins anddoped carriers tend to spatially separate into 1- or 2-D stripes. Some viewthese as central to superconductivity, others as peripheral and merelycompeting. Using La$_{2-x}$Sr$_x$Cu$_{1-y}$Zn$_y$O$_4$ we show that an oxygenisotope effect in $T_c$ and in the superfluid density can be used todistinguish between the roles of stripes and pseudogap and also to detect thepresence of impurity scattering. We conclude that stripes and pseudogap aredistinct, and both compete and coexist with superconductivity.Comment: Revised submission to PRL with added appendix on a possible isotope  effect in the effective mass, 4 pages, 3 figure

Isotope Effect in the Superfluid Density of High-Temperature Superconducting Cuprates: Stripes, Pseudogap, and Impurities