Question
Will vertical 4" cracks all around the bark at the base of the trunk kill the tree?  
This is  tree is relatively young, about 20' tall, and planted in Los Angeles. Thank
you for any info.

Answer
Sycamore tree tend to shed their bark as they get older so it sounds like the "cracks" are just the bark splitting as it grows and peels. This starts near the base of the tree. As long as the foliage is green and full there is nothing to worry about. The bark over time will peel do not try to help it by peeling the loose bark off this could damage the tree. Sounds normal to me.