If you're not going to go see a doctor, then you can try the following: Trim the dead skin from the edges, squirt some Desitin into the crack and butterfly it shut. Pure Lanolin also works well, and can also be found in the baby aisle at most pharmacys (Nursing mothers use it for cracked nipples). Or just cream up with the Desitin and cover it with a band-aid.
Alternately, you can trim the dead skin from the edges and super glue the crack shut.
Either way, I would recommend using a heavy duty moisturizer like Bag Balm (which also has lanolin) every night at bedtime (best to wear socks on hands or gloves if you don't want greasy sheets). And drink lots of water.
If it doesn't get better it could be a fungus, in which case see a doctor.
Best regards,
Bruce Mitchell