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Topology"],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Given a relation\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$R \\subseteq I \\times J$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo>\u2286<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>J<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    between two sets, Dowker\u2019s Theorem (1952) states that the homology groups of two associated simplicial complexes\u2014now known as Dowker complexes\u2014are isomorphic. In its modern form, the full result asserts a functorial homotopy equivalence between the two Dowker complexes. What can be said about relations defined on three or more sets? We present a simple generalization to \u2018multiway\u2019 relations of the form\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$R \\subseteq I_1 \\times I_2 \\times \\cdots \\times I_m$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo>\u2286<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>I<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>I<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mo>\u22ef<\/mml:mo>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>I<\/mml:mi>\n                              <mml:mi>m<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . The theorem asserts functorial homotopy equivalences between\n                    <jats:italic>m<\/jats:italic>\n                    \u00a0multiway Dowker complexes and a variant of the rectangle complex of Brun and Salbu from their recent short proof of Dowker\u2019s Theorem. Our proof uses Smale\u2019s homotopy mapping theorem and factors through a \u2018cellular Dowker lemma\u2019 that expresses the main idea in more general form. To make the geometry more transparent, we work with a class of spaces called \u2018prod-complexes\u2019 then transfer the results to simplicial complexes through a \u2018simplexification\u2019 process. We conclude with a detailed study of ternary relations, identifying seven functorially defined homotopy types and twelve natural transformations between them.\n                  <\/jats:p>","DOI":"10.1007\/s41468-026-00239-x","type":"journal-article","created":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T12:26:35Z","timestamp":1780921595000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dowker\u2019s theorem for higher-order relations"],"prefix":"10.1007","volume":"10","author":[{"given":"Vin","family":"de Silva","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chad","family":"Giusti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vladimir","family":"Itskov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Robinson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Radmila","family":"Sazdanovic","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nikolas","family":"Schonsheck","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Melvin","family":"Vaupel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Iris","family":"Yoon","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,8]]},"reference":[{"issue":"3","key":"239_CR1","doi-asserted-by":"publisher","first-page":"534","DOI":"10.2307\/1969366","volume":"51","author":"EG Begle","year":"1950","unstructured":"Begle, E.G.: The Vietoris mapping theorem for bicompact spaces. 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