DSPACE(n)=<SUP>2</SUP> NSPACE(n): a degree theoretic characterization

It is shown that the following are equivalent. 1. DSPACE(n)=NSPACE(n). 2. There is a nontrivial &#8804; 1-NLm-degree that coincides with &#8804;1-Lm-degree. 3. For every class C closed under log-lin reductions, the &#8804;1-NLm-complete degree of C coincides with the &#8804;1-Lm-complete degree of C