Provided are a method, system, and computer readable storage medium for managing access to a storage device. A logical-to-physical mapping indicates for each logical address a physical address in the storage device having current data for the logical address and version information indicating whether there is a prior version of data for the logical address. In response to the logical-to-physical mapping indicating that there is no prior version of the data for a target logical address of a write, including information on the target physical address and the physical address indicated in the logical-to-physical mapping in checkpoint information. The version information for the target logical address is updated to indicate that there is a prior version of data. Data for the write is written to a target physical address. The logical-to-physical mapping for the target logical address is updated to indicate the target physical address.