Sadofsky, HalMerrill, Leanne2018-04-102018-04-102018-04-10https://hdl.handle.net/1794/23144The Periodicity theorem of Hopkins and Smith tells us that any finite spectrum supports a $v_n$-map for some $n$. We are interested in finding finite $2$-local spectra that both support a $v_2$-map with a low power of $v_2$ and have few cells. Following the process outlined in Palmieri-Sadofsky, we study a related class of self-maps, known as $u_2$-maps, between stably finite spectra. We construct examples of spectra that might be expected to support $u_2^1$-maps, and then we use Margolis homology and homological algebra computations to show that they do not support $u_2^1$-maps. We also show that one example does not support a $u_2^2$-map. The nonexistence of $u_2$-maps on these spectra eliminates certain examples from consideration by this technique.en-USAll Rights Reserved.Algebraic topologyHomotopy theoryElectronic Thesis or Dissertation