It has been known that the configuration space F(R2, n) of n distinct ordered points in R2 deformation retracts to a regular CW complex with n!permutohedra Pn as the top dimensional cells. In this paper, we show that there exists a similar but different permutohedral structure of the spaceCact(n) of spineless cacti with n lobes. Based on these structures, direct homotopy equivalences between F (R2, n) and Cact(n) are then given. It is well known that the little 2-discs space D2(n) is homotopy equivalent toF(R2, n). Our results give partial combinatorial and geometrical interpretation of the equivalences between D2 and Cact.
","abstract_html":"<p>It has been known that the configuration space F(R<sup>2</sup>, <em>n)</em> of <em>n</em> distinct ordered points in R<sup>2</sup> deformation retracts to a regular CW complex with <em>n!</em>permutohedra <em>P<sub>n</sub></em> as the top dimensional cells. In this paper, we show that there exists a similar but different permutohedral structure of the space<em>Cact(n)</em> of spineless cacti with n lobes. Based on these structures, direct homotopy equivalences between <em>F</em> (R<sup>2</sup>, <em>n)</em> and <em>Cact(n)</em> are then given. It is well known that the little 2-discs space <em>D<sub>2</sub>(n)</em> is homotopy equivalent to<em>F</em>(R<sup>2</sup>, <em>n).</em> Our results give partial combinatorial and geometrical interpretation of the equivalences between <em>D<sub>2</sub></em> and <em>Cact.</em></p>","abstract_has_math":false,"creators":["Zhang, Yongheng"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ralph M. Kaufmann","James McClure","David B. McReynolds","David Gepner"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-04-01T07:00:00Z","date_published":"2015-04-01T07:00:00Z","updated_at":"2026-07-24T03:53:41Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://docs.lib.purdue.edu/open_access_dissertations/607","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ralph M. Kaufmann","James McClure","David B. McReynolds","David Gepner"]},{"key":"dc:creator","label":"Author","values":["Zhang, Yongheng"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://docs.lib.purdue.edu/open_access_dissertations/607"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["It has been known that the configuration space F(R2, n) of n distinct ordered points in R2 deformation retracts to a regular CW complex with n!permutohedra Pn as the top dimensional cells. In this paper, we show that there exists a similar but different permutohedral structure of the spaceCact(n) of spineless cacti with n lobes. Based on these structures, direct homotopy equivalences between F (R2, n) and Cact(n) are then given. It is well known that the little 2-discs space D2(n) is homotopy equivalent toF(R2, n). Our results give partial combinatorial and geometrical interpretation of the equivalences between D2 and Cact.
"]},{"key":"dc:title","label":"Title","values":["Permutohedra, configuration spaces and spineless cacti"]}]}],"canonical_facts":{"dc:contributor":["Ralph M. Kaufmann","James McClure","David B. McReynolds","David Gepner"],"dc:creator":["Zhang, Yongheng"],"dc:description.abstract":["It has been known that the configuration space F(R2, n) of n distinct ordered points in R2 deformation retracts to a regular CW complex with n!permutohedra Pn as the top dimensional cells. In this paper, we show that there exists a similar but different permutohedral structure of the spaceCact(n) of spineless cacti with n lobes. Based on these structures, direct homotopy equivalences between F (R2, n) and Cact(n) are then given. It is well known that the little 2-discs space D2(n) is homotopy equivalent toF(R2, n). Our results give partial combinatorial and geometrical interpretation of the equivalences between D2 and Cact.
"],"dc:identifier":["https://docs.lib.purdue.edu/open_access_dissertations/607"],"dc:subject":["Mathematics"],"dc:title":["Permutohedra, configuration spaces and spineless cacti"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:53:41Z"}