{"id":{"repo_id":"umn","oai_identifier":"oai:conservancy.umn.edu:11299/241394"},"canonical_url":"https://search.dev.ndltd.org/etd/umn/oai:conservancy.umn.edu:11299/241394","repository":{"repo_id":"umn","name":"University of Minnesota","base_url":"https://conservancy.umn.edu/server/oai/request"},"display":{"title":"On the Power Operations of MU","abstract":"We study the power opearations of the spectrum $MU$, the complex cobordism theory. After reviewing necessary backgrounds, we start by recalling a formula from Justin Noel and Niles Johnson connecting the power operation $P([\\mb{CP}^n])$ for $[\\mb{CP}^n]\\in MU^*$ with the power operation $P_{\\mb{CP}}(x)$ for the orientation class $x\\in MU^*(\\mb{CP}^\\infty)$. Next, we find an algorithm calculating the effect of $P$ on a set of polynomial generators $x_i\\in MU^*$ from the known formulas coverting $[\\mb{CP}^n]$'s to $x_i$'s and the fact that under the canonical projection $q:MU^*\\bbra{\\alpha}/\\bra{p}\\rightarrow MU^*\\bbra{\\alpha}/\\ang{p},$ both $q_*P$ and $q_*P_{\\mb{CP}}$ become ring homomorphisms. Finally, we display a sample calculation of $P(x_3)$ at $p=2$ with the help of Maple, and provide an application of our calculation where we put some restrictions on possible $E_3$ maps from $MU$ to $BP$(or $BP\\langle n\\rangle$'s).","abstract_html":"We study the power opearations of the spectrum $MU$, the complex cobordism theory. After reviewing necessary backgrounds, we start by recalling a formula from Justin Noel and Niles Johnson connecting the power operation <span class=\"etd-inline-math\">P([\\mb{CP}<sup>n</sup>])</span> for <span class=\"etd-inline-math\">[\\mb{CP}<sup>n</sup>]\\in MU<sup>*</sup></span> with the power operation <span class=\"etd-inline-math\">P<sub>\\mb{CP}</sub>(x)</span> for the orientation class <span class=\"etd-inline-math\">x\\in MU<sup>*</sup>(\\mb{CP}<sup>\\</sup>infty)</span>. Next, we find an algorithm calculating the effect of $P$ on a set of polynomial generators <span class=\"etd-inline-math\">x<sub>i</sub>\\in MU<sup>*</sup></span> from the known formulas coverting <span class=\"etd-inline-math\">[\\mb{CP}<sup>n</sup>]</span>&#x27;s to <span class=\"etd-inline-math\">x<sub>i</sub></span>&#x27;s and the fact that under the canonical projection <span class=\"etd-inline-math\">q:MU<sup>*</sup>\\bbra{&alpha;}/\\bra{p}\\rightarrow MU<sup>*</sup>\\bbra{&alpha;}/\\ang{p},</span> both <span class=\"etd-inline-math\">q<sub>*</sub>P</span> and <span class=\"etd-inline-math\">q<sub>*</sub>P<sub>\\mb{CP}</sub></span> become ring homomorphisms. Finally, we display a sample calculation of <span class=\"etd-inline-math\">P(x<sub>3</sub>)</span> at $p=2$ with the help of Maple, and provide an application of our calculation where we put some restrictions on possible <span class=\"etd-inline-math\">E<sub>3</sub></span> maps from $MU$ to $BP$(or $BP\\langle n\\rangle$&#x27;s).","abstract_has_math":true,"creators":["Gu, Zeshen"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-04","date_published":"2022-04","updated_at":"2026-07-24T05:19:46Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/11299/241394","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Gu, Zeshen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-08-29T19:24:17Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-08-29T19:24:17Z"]},{"key":"dc:date.issued","label":"Date","values":["2022-04"]},{"key":"dc:type","label":"Dc Type","values":["Thesis or Dissertation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/11299/241394"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["University of Minnesota Ph.D. dissertation. April 2022. Major: Mathematics. Advisor: Tyler Lawson. 1 computer file (PDF); ii, 54 pages."]},{"key":"dc:description.abstract","label":"Abstract","values":["We study the power opearations of the spectrum $MU$, the complex cobordism theory. After reviewing necessary backgrounds, we start by recalling a formula from Justin Noel and Niles Johnson connecting the power operation $P([\\mb{CP}^n])$ for $[\\mb{CP}^n]\\in MU^*$ with the power operation $P_{\\mb{CP}}(x)$ for the orientation class $x\\in MU^*(\\mb{CP}^\\infty)$. Next, we find an algorithm calculating the effect of $P$ on a set of polynomial generators $x_i\\in MU^*$ from the known formulas coverting $[\\mb{CP}^n]$'s to $x_i$'s and the fact that under the canonical projection $q:MU^*\\bbra{\\alpha}/\\bra{p}\\rightarrow MU^*\\bbra{\\alpha}/\\ang{p},$ both $q_*P$ and $q_*P_{\\mb{CP}}$ become ring homomorphisms. Finally, we display a sample calculation of $P(x_3)$ at $p=2$ with the help of Maple, and provide an application of our calculation where we put some restrictions on possible $E_3$ maps from $MU$ to $BP$(or $BP\\langle n\\rangle$'s)."]},{"key":"dc:title","label":"Title","values":["On the Power Operations of MU"]}]}],"canonical_facts":{"dc:creator":["Gu, Zeshen"],"dc:date.accessioned":["2022-08-29T19:24:17Z"],"dc:date.available":["2022-08-29T19:24:17Z"],"dc:date.issued":["2022-04"],"dc:description":["University of Minnesota Ph.D. dissertation. April 2022. Major: Mathematics. Advisor: Tyler Lawson. 1 computer file (PDF); ii, 54 pages."],"dc:description.abstract":["We study the power opearations of the spectrum $MU$, the complex cobordism theory. After reviewing necessary backgrounds, we start by recalling a formula from Justin Noel and Niles Johnson connecting the power operation $P([\\mb{CP}^n])$ for $[\\mb{CP}^n]\\in MU^*$ with the power operation $P_{\\mb{CP}}(x)$ for the orientation class $x\\in MU^*(\\mb{CP}^\\infty)$. Next, we find an algorithm calculating the effect of $P$ on a set of polynomial generators $x_i\\in MU^*$ from the known formulas coverting $[\\mb{CP}^n]$'s to $x_i$'s and the fact that under the canonical projection $q:MU^*\\bbra{\\alpha}/\\bra{p}\\rightarrow MU^*\\bbra{\\alpha}/\\ang{p},$ both $q_*P$ and $q_*P_{\\mb{CP}}$ become ring homomorphisms. Finally, we display a sample calculation of $P(x_3)$ at $p=2$ with the help of Maple, and provide an application of our calculation where we put some restrictions on possible $E_3$ maps from $MU$ to $BP$(or $BP\\langle n\\rangle$'s)."],"dc:identifier.uri":["https://hdl.handle.net/11299/241394"],"dc:language.iso":["en"],"dc:title":["On the Power Operations of MU"],"dc:type":["Thesis or Dissertation"]},"updated_at":"2026-07-24T05:19:46Z"}