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Topology"],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let\u00a0<jats:italic>f<\/jats:italic> be a Morse function on a smooth compact manifold\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$M$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with boundary. The path component\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathrm {PH}^{-1}_f(D)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mrow>\n                        <mml:mi>PH<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msubsup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>D<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> containing\u00a0<jats:italic>f<\/jats:italic> of the space of Morse functions giving rise to the same Persistent Homology\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$D=\\mathrm {PH}(f)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>D<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>PH<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is shown to be the same as the orbit of\u00a0<jats:italic>f<\/jats:italic> under pre-composition\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi \\mapsto f\\circ \\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03d5<\/mml:mi>\n                    <mml:mo>\u21a6<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>\u2218<\/mml:mo>\n                    <mml:mi>\u03d5<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> by diffeomorphisms of\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$M$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> which are isotopic to the identity. Consequently we derive topological properties of the fiber\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathrm {PH}^{-1}_f(D)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mrow>\n                        <mml:mi>PH<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msubsup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>D<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>: In particular we compute its homotopy type for many compact surfaces\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$M$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In the\u00a01-dimensional settings where\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$M$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the unit interval or the circle we extend the analysis to continuous functions and show that the fibers are made of contractible and circular components respectively.\n<\/jats:p>","DOI":"10.1007\/s41468-022-00100-x","type":"journal-article","created":{"date-parts":[[2022,10,10]],"date-time":"2022-10-10T12:26:59Z","timestamp":1665404819000},"page":"89-102","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Fiber of persistent homology on morse functions"],"prefix":"10.1007","volume":"7","author":[{"given":"Jacob","family":"Leygonie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Beers","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,10,10]]},"reference":[{"key":"100_CR1","unstructured":"Barannikov, S.: The framed Morse complex and its invariants. 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