Would a sparse NP-Complete language imply L = NP?
Update: Thanks to Noah Schweber for clear and comprehensive answer. Having thought about it more, one would need a logspace reduction from NP-complete to L-complete for NP=L.  below does not directly address this.
Here is my proof attempt:
 Assume there exists a (sparse NP-Complete language).
Mahaney's theorem is a theorem in computational complexity theory proven by Stephen Mahaney that states that if any sparse language is NP-Complete, then P=NP.
 In other words, (sparse NP-Complete language) implies (P = NP).
 From  and  (P = NP).
 (P = NP) implies (P-complete = NP-complete).
 From  and  (P-complete = NP-complete).
 From  and  there exists a (sparse P-Complete language)
In 1999, Jin-Yi Cai and D. Sivakumar, building on work by Ogihara, showed that if there exists a sparse P-complete problem, then L = P."
 In other words, (sparse P-complete problem) implies (L = P).
 From  and  (L = P).
 From  and  (L = NP).