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Hopf surface
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<h2 id="invariants">Invariants</h2> <p>Hopf surfaces are <a href="/facts/Surfaces_of_class_VII/YnJl2vud">surfaces of class VII</a> and in particular all have <a href="/facts/Kodaira_dimension/dkG0mUuq">Kodaira dimension</a> − ∞ {\displaystyle -\infty } , and all their plurigenera vanish. The geometric genus is 0. The <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> has a normal central infinite cyclic subgroup of finite index. The <a href="/facts/Homological_mirror_symmetry/Wbv35FAn">Hodge diamond</a> is </p> <table><tbody><tr><td></td><td></td><td>1</td><td></td><td></td></tr><tr><td></td><td>0</td><td></td><td>1</td><td></td></tr><tr><td>0</td><td></td><td>0</td><td></td><td>0</td></tr><tr><td></td><td>1</td><td></td><td>0</td><td></td></tr><tr><td></td><td></td><td>1</td><td></td><td></td></tr></tbody></table> <p>In particular the first <a href="/facts/Betti_number/r42Y8k6f">Betti number</a> is 1 and the second Betti number is 0. Conversely <a href="/facts/Kunihiko_Kodaira/u73B5W0D">Kunihiko Kodaira</a> (1968) showed that a compact complex surface with vanishing the second Betti number and whose fundamental group contains an infinite cyclic subgroup of finite index is a Hopf surface. </p> <h2 id="primary-hopf-surfaces">Primary Hopf surfaces</h2> <p>In the course of <a href="/facts/Enriques%25E2%2580%2593Kodaira_classification/Lr1rd0Nf">classification of compact complex surfaces</a>, Kodaira classified the primary Hopf surfaces. </p><p>A primary Hopf surface is obtained as </p> H = ( C 2 ∖ { 0 } ) / Γ , {\displaystyle H={\Big (}\mathbb {C} ^{2}\setminus \{0\}{\Big )}/\Gamma ,} <p>where Γ {\displaystyle \Gamma } is a group generated by a polynomial contraction γ {\displaystyle \gamma } . Kodaira has found a normal form for γ {\displaystyle \gamma } . In appropriate coordinates, γ {\displaystyle \gamma } can be written as </p> ( x , y ) ↦ ( α x + λ y n , β y ) {\displaystyle (x,y)\mapsto (\alpha x+\lambda y^{n},\beta y)} <p>where α , β ∈ C {\displaystyle \alpha ,\beta \in \mathbb {C} } are <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> satisfying 0 < | α | ≤ | β | < 1 {\displaystyle 0<|\alpha |\leq |\beta |<1} , and either λ = 0 {\displaystyle \lambda =0} or α = β n {\displaystyle \alpha =\beta ^{n}} . </p><p>These surfaces contain an elliptic curve (the image of the <i>x</i>-axis) and if λ = 0 {\displaystyle \lambda =0} the image of the <i>y</i>-axis is a second elliptic curve. When λ = 0 {\displaystyle \lambda =0} , the Hopf surface is an elliptic fiber space over the projective line if α m = β n {\displaystyle \alpha ^{m}=\beta ^{n}} for some positive integers <i>m</i> and <i>n</i>, with the map to the projective line given by ( x , y ) ↦ x m y − n {\displaystyle (x,y)\mapsto x^{m}y^{-n}} , and otherwise the only curves are the two images of the axes. </p><p>The <a href="/facts/Picard_group/lWH0CoWQ">Picard group</a> of any primary Hopf surface is isomorphic to the non-zero complex numbers C ∗ {\displaystyle \mathbb {C} ^{*}} . </p><p>Kodaira (1966b) has proven that a complex surface is diffeomorphic to S 3 × S 1 {\displaystyle S^{3}\times S^{1}} if and only if it is a primary Hopf surface. </p> <h2 id="secondary-hopf-surfaces">Secondary Hopf surfaces</h2> <p>Any secondary Hopf surface has a finite unramified cover that is a primary Hopf surface. Equivalently, its fundamental group has a subgroup of finite index in its center that is isomorphic to the integers. Masahido Kato (1975) classified them by finding the finite groups acting without fixed points on primary Hopf surfaces. </p><p>Many examples of secondary Hopf surfaces can be constructed with underlying space a product of a <a href="/facts/Spherical_space_form/8zJWCqh9">spherical space forms</a> and a circle. </p> <ul><li>Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), <i>Compact Complex Surfaces</i>, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-57739-0">10.1007/978-3-642-57739-0</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-00832-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2030225">2030225</a></li> <li><a href="/facts/Heinz_Hopf/PMks10NV">Hopf, Heinz</a> (1948). "Zur Topologie der komplexen Mannigfaltigkeiten". <i>Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948</i>. Interscience Publishers, Inc., New York. pp. 167–185. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0023054">0023054</a>.</li> <li>Kato, Masahide (1975), "Topology of Hopf surfaces", <i>Journal of the Mathematical Society of Japan</i>, 27 (2): 222–238, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2969%2Fjmsj%2F02720222">10.2969/jmsj/02720222</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5645">0025-5645</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0402128">0402128</a> Kato, Masahide (1989), "Erratum to: "Topology of Hopf surfaces"", <i>Journal of the Mathematical Society of Japan</i>, 41 (1): 173–174, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2969%2Fjmsj%2F04110173">10.2969/jmsj/04110173</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5645">0025-5645</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0972171">0972171</a></li> <li><a href="/facts/Kunihiko_Kodaira/u73B5W0D">Kodaira, Kunihiko</a> (1966), "On the structure of compact complex analytic surfaces. II", <i><a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a></i>, 88 (3), The Johns Hopkins University Press: 682–721, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2373150">10.2307/2373150</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2373150">2373150</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0205280">0205280</a>, <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC300219">300219</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/16578569">16578569</a></li> <li><a href="/facts/Kunihiko_Kodaira/u73B5W0D">Kodaira, Kunihiko</a> (1968), "On the structure of compact complex analytic surfaces. III", <i><a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a></i>, 90 (1), The Johns Hopkins University Press: 55–83, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2373426">10.2307/2373426</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2373426">2373426</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0228019">0228019</a></li> <li>Kodaira, Kunihiko (1966b), <a href="http://www.pnas.org/content/55/2/240.full.pdf+html">"Complex structures on S1×S3"</a>, <i><a href="/facts/Proceedings_of_the_National_Academy_of_Sciences/w75CThX0">Proceedings of the National Academy of Sciences of the United States of America</a></i>, 55 (2): 240–243, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1966PNAS...55..240K">1966PNAS...55..240K</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1073%2Fpnas.55.2.240">10.1073/pnas.55.2.240</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0196769">0196769</a>, <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC224129">224129</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/16591329">16591329</a></li> <li>Matumoto, Takao; Nakagawa, Noriaki (2000), <a href="http://projecteuclid.org/euclid.ojm/1200789206">"Explicit description of Hopf surfaces and their automorphism groups"</a>, <i>Osaka Journal of Mathematics</i>, 37 (2): 417–424, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0030-6126">0030-6126</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1772841">1772841</a></li> <li>Ornea, Liviu (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Hopf_manifold">"Hopf manifold"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>