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Genus of a multiplicative sequence
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<h2 id="definition">Definition</h2> <p>A genus φ {\displaystyle \varphi } assigns a number Φ ( X ) {\displaystyle \Phi (X)} to each manifold <i>X</i> such that </p> <ol><li> Φ ( X ⊔ Y ) = Φ ( X ) + Φ ( Y ) {\displaystyle \Phi (X\sqcup Y)=\Phi (X)+\Phi (Y)} (where ⊔ {\displaystyle \sqcup } is the disjoint union);</li> <li> Φ ( X × Y ) = Φ ( X ) Φ ( Y ) {\displaystyle \Phi (X\times Y)=\Phi (X)\Phi (Y)} ;</li> <li> Φ ( X ) = 0 {\displaystyle \Phi (X)=0} if <i>X</i> is the boundary of a manifold with boundary.</li></ol> <p>The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see <a href="/facts/List_of_cohomology_theories/MkqxXM9M">list of cobordism theories</a> for many more examples). The value Φ ( X ) {\displaystyle \Phi (X)} is in some ring, often the ring of rational numbers, though it can be other rings such as Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } or the ring of modular forms. </p><p>The conditions on Φ {\displaystyle \Phi } can be rephrased as saying that φ {\displaystyle \varphi } is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring. </p><p>Example: If Φ ( X ) {\displaystyle \Phi (X)} is the <a href="/facts/Signature_(topology)/n76t7Mjf">signature</a> of the oriented manifold <i>X</i>, then Φ {\displaystyle \Phi } is a genus from oriented manifolds to the ring of integers. </p> <h2 id="the-genus-associated-to-a-formal-power-series">The genus associated to a formal power series</h2> <p class="note">Main article: <a href="/facts/Multiplicative_sequence/mkXy0zWl">Multiplicative sequence</a></p> <p>A sequence of polynomials K 1 , K 2 , … {\displaystyle K_{1},K_{2},\ldots } in variables p 1 , p 2 , … {\displaystyle p_{1},p_{2},\ldots } is called multiplicative if </p> 1 + p 1 z + p 2 z 2 + ⋯ = ( 1 + q 1 z + q 2 z 2 + ⋯ ) ( 1 + r 1 z + r 2 z 2 + ⋯ ) {\displaystyle 1+p_{1}z+p_{2}z^{2}+\cdots =(1+q_{1}z+q_{2}z^{2}+\cdots )(1+r_{1}z+r_{2}z^{2}+\cdots )} <p>implies that </p> ∑ j K j ( p 1 , p 2 , … ) z j = ∑ j K j ( q 1 , q 2 , … ) z j ∑ k K k ( r 1 , r 2 , … ) z k {\displaystyle \sum _{j}K_{j}(p_{1},p_{2},\ldots )z^{j}=\sum _{j}K_{j}(q_{1},q_{2},\ldots )z^{j}\sum _{k}K_{k}(r_{1},r_{2},\ldots )z^{k}} <p>If Q ( z ) {\displaystyle Q(z)} is a <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> in <i>z</i> with constant term 1, we can define a multiplicative sequence </p> K = 1 + K 1 + K 2 + ⋯ {\displaystyle K=1+K_{1}+K_{2}+\cdots } <p>by </p> K ( p 1 , p 2 , p 3 , … ) = Q ( z 1 ) Q ( z 2 ) Q ( z 3 ) ⋯ {\displaystyle K(p_{1},p_{2},p_{3},\ldots )=Q(z_{1})Q(z_{2})Q(z_{3})\cdots } , <p>where p k {\displaystyle p_{k}} is the <i>k</i>th <a href="/facts/Elementary_symmetric_function/9RFXe9b6">elementary symmetric function</a> of the indeterminates z i {\displaystyle z_{i}} . (The variables p k {\displaystyle p_{k}} will often in practice be <a href="/facts/Pontryagin_class/TIFceC5j">Pontryagin classes</a>.) </p><p>The genus Φ {\displaystyle \Phi } of <a href="/facts/Compact_space/d0cgXJH7">compact</a>, <a href="/facts/Connected_space/5CUTxfoA">connected</a>, <a href="/facts/Differentiable_manifold/aSnbXKcV">smooth</a>, <a href="/facts/Orientability/1SEQM2Pt">oriented</a> manifolds corresponding to <i>Q</i> is given by </p> Φ ( X ) = K ( p 1 , p 2 , p 3 , … ) {\displaystyle \Phi (X)=K(p_{1},p_{2},p_{3},\ldots )} <p>where the p k {\displaystyle p_{k}} are the <a href="/facts/Pontryagin_class/TIFceC5j">Pontryagin classes</a> of <i>X</i>. The power series <i>Q</i> is called the characteristic power series of the genus Φ {\displaystyle \Phi } . A theorem of <a href="/facts/Ren%25C3%25A9_Thom/hdzhWSaC">René Thom</a>, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4<i>k</i> for positive integers <i>k</i>, implies that this gives a bijection between formal power series <i>Q</i> with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers. </p> <h2 id="l-genus">L genus</h2> <p>The L genus is the genus of the formal power series </p> z tanh ( z ) = ∑ k ≥ 0 2 2 k B 2 k z k ( 2 k ) ! = 1 + z 3 − z 2 45 + ⋯ {\displaystyle {{\sqrt {z}} \over \tanh({\sqrt {z}})}=\sum _{k\geq 0}{\frac {2^{2k}B_{2k}z^{k}}{(2k)!}}=1+{z \over 3}-{z^{2} \over 45}+\cdots } <p>where the numbers B 2 k {\displaystyle B_{2k}} are the <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli numbers</a>. The first few values are: </p> L 0 = 1 L 1 = 1 3 p 1 L 2 = 1 45 ( 7 p 2 − p 1 2 ) L 3 = 1 945 ( 62 p 3 − 13 p 1 p 2 + 2 p 1 3 ) L 4 = 1 14175 ( 381 p 4 − 71 p 1 p 3 − 19 p 2 2 + 22 p 1 2 p 2 − 3 p 1 4 ) {\displaystyle {\begin{aligned}L_{0}&=1\\L_{1}&={\tfrac {1}{3}}p_{1}\\L_{2}&={\tfrac {1}{45}}\left(7p_{2}-p_{1}^{2}\right)\\L_{3}&={\tfrac {1}{945}}\left(62p_{3}-13p_{1}p_{2}+2p_{1}^{3}\right)\\L_{4}&={\tfrac {1}{14175}}\left(381p_{4}-71p_{1}p_{3}-19p_{2}^{2}+22p_{1}^{2}p_{2}-3p_{1}^{4}\right)\end{aligned}}} <p>(for further <i>L</i>-polynomials see <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> or <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A237111). Now let <i>M</i> be a closed smooth oriented manifold of dimension 4<i>n</i> with <a href="/facts/Pontrjagin_class/TIFceC5j">Pontrjagin classes</a> p i = p i ( M ) {\displaystyle p_{i}=p_{i}(M)} . <a href="/facts/Friedrich_Hirzebruch/5MMEInjc">Friedrich Hirzebruch</a> showed that the <i>L</i> genus of <i>M</i> in dimension 4<i>n</i> evaluated on the <a href="/facts/Fundamental_class/bqmdWEEc">fundamental class</a> of M {\displaystyle M} , denoted [ M ] {\displaystyle [M]} , is equal to σ ( M ) {\displaystyle \sigma (M)} , the <a href="/facts/Signature_(topology)/n76t7Mjf">signature</a> of <i>M</i> (i.e., the signature of the <a href="/facts/Intersection_theory/UgckwVGF">intersection form</a> on the 2<i>n</i>th cohomology group of <i>M</i>): </p> σ ( M ) = ⟨ L n ( p 1 ( M ) , … , p n ( M ) ) , [ M ] ⟩ {\displaystyle \sigma (M)=\langle L_{n}(p_{1}(M),\ldots ,p_{n}(M)),[M]\rangle } . <p>This is now known as the <a href="/facts/Hirzebruch_signature_theorem/uDcIGALT">Hirzebruch signature theorem</a> (or sometimes the Hirzebruch index theorem). </p><p>The fact that L 2 {\displaystyle L_{2}} is always integral for a smooth manifold was used by <a href="/facts/John_Milnor/LTf24paN">John Milnor</a> to give an example of an 8-dimensional <a href="/facts/PL_manifold/dq3fTkPT">PL manifold</a> with no <a href="/facts/Smooth_structure/lfvfhqxr">smooth structure</a>. Pontryagin numbers can also be defined for PL manifolds, and Milnor showed that his PL manifold had a non-integral value of p 2 {\displaystyle p_{2}} , and so was not smoothable. </p> <h3>Application on K3 surfaces</h3> <p>Since projective <a href="/facts/Pontryagin_class/TIFceC5j">K3 surfaces</a> are smooth complex manifolds of dimension two, their only non-trivial <a href="/facts/Pontryagin_class/TIFceC5j">Pontryagin class</a> is p 1 {\displaystyle p_{1}} in H 4 ( X ) {\displaystyle H^{4}(X)} . It can be computed as -48 using the tangent sequence and comparisons with complex chern classes. Since L 1 = − 16 {\displaystyle L_{1}=-16} , we have its signature. This can be used to compute its intersection form as a unimodular lattice since it has dim ( H 2 ( X ) ) = 22 {\displaystyle \operatorname {dim} \left(H^{2}(X)\right)=22} , and using the classification of unimodular lattices.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="todd-genus">Todd genus</h2> <p>The Todd genus is the genus of the formal power series </p> z 1 − exp ( − z ) = ∑ i = 0 ∞ B i i ! z i {\displaystyle {\frac {z}{1-\exp(-z)}}=\sum _{i=0}^{\infty }{\frac {B_{i}}{i!}}z^{i}} <p>with B i {\displaystyle B_{i}} as before, Bernoulli numbers. The first few values are </p> T d 0 = 1 T d 1 = 1 2 c 1 T d 2 = 1 12 ( c 2 + c 1 2 ) T d 3 = 1 24 c 1 c 2 T d 4 = 1 720 ( − c 1 4 + 4 c 2 c 1 2 + 3 c 2 2 + c 3 c 1 − c 4 ) {\displaystyle {\begin{aligned}Td_{0}&=1\\Td_{1}&={\frac {1}{2}}c_{1}\\Td_{2}&={\frac {1}{12}}\left(c_{2}+c_{1}^{2}\right)\\Td_{3}&={\frac {1}{24}}c_{1}c_{2}\\Td_{4}&={\frac {1}{720}}\left(-c_{1}^{4}+4c_{2}c_{1}^{2}+3c_{2}^{2}+c_{3}c_{1}-c_{4}\right)\end{aligned}}} <p>The Todd genus has the particular property that it assigns the value 1 to all complex projective spaces (i.e. T d n ( C P n ) = 1 {\displaystyle \mathrm {Td} _{n}(\mathbb {CP} ^{n})=1} ), and this suffices to show that the Todd genus agrees with the arithmetic genus for algebraic varieties as the <a href="/facts/Arithmetic_genus/hYdDY8Su">arithmetic genus</a> is also 1 for complex projective spaces. This observation is a consequence of the <a href="/facts/Hirzebruch%25E2%2580%2593Riemann%25E2%2580%2593Roch_theorem/PxsJ0BSm">Hirzebruch–Riemann–Roch theorem</a>, and in fact is one of the key developments that led to the formulation of that theorem. </p> <h2 id="-genus"> genus</h2> <p>The  genus is the genus associated to the characteristic power series </p> Q ( z ) = 1 2 z sinh ( 1 2 z ) = 1 − z 24 + 7 z 2 5760 − ⋯ {\displaystyle Q(z)={\frac {{\frac {1}{2}}{\sqrt {z}}}{\sinh \left({\frac {1}{2}}{\sqrt {z}}\right)}}=1-{\frac {z}{24}}+{\frac {7z^{2}}{5760}}-\cdots } <p>(There is also an A genus which is less commonly used, associated to the characteristic series Q ( 16 z ) {\displaystyle Q(16z)} .) The first few values are </p> A ^ 0 = 1 A ^ 1 = − 1 24 p 1 A ^ 2 = 1 5760 ( − 4 p 2 + 7 p 1 2 ) A ^ 3 = 1 967680 ( − 16 p 3 + 44 p 2 p 1 − 31 p 1 3 ) A ^ 4 = 1 464486400 ( − 192 p 4 + 512 p 3 p 1 + 208 p 2 2 − 904 p 2 p 1 2 + 381 p 1 4 ) {\displaystyle {\begin{aligned}{\hat {A}}_{0}&=1\\{\hat {A}}_{1}&=-{\tfrac {1}{24}}p_{1}\\{\hat {A}}_{2}&={\tfrac {1}{5760}}\left(-4p_{2}+7p_{1}^{2}\right)\\{\hat {A}}_{3}&={\tfrac {1}{967680}}\left(-16p_{3}+44p_{2}p_{1}-31p_{1}^{3}\right)\\{\hat {A}}_{4}&={\tfrac {1}{464486400}}\left(-192p_{4}+512p_{3}p_{1}+208p_{2}^{2}-904p_{2}p_{1}^{2}+381p_{1}^{4}\right)\end{aligned}}} <p>The  genus of a <a href="/facts/Spin_manifold/BF78kFz8">spin manifold</a> is an integer, and an even integer if the dimension is 4 mod 8 (which in dimension 4 implies <a href="/facts/Rochlin%2527s_theorem/lwaxv5lW">Rochlin's theorem</a>) – for general manifolds, the  genus is not always an integer. This was proven by <a href="/facts/Friedrich_Hirzebruch/5MMEInjc">Hirzebruch</a> and <a href="/facts/Armand_Borel/QkRUS4ej">Armand Borel</a>; this result both motivated and was later explained by the <a href="/facts/Atiyah%25E2%2580%2593Singer_index_theorem/qwcSxamI">Atiyah–Singer index theorem</a>, which showed that the  genus of a spin manifold is equal to the index of its <a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a>. </p><p>By combining this index result with a <a href="/facts/Weitzenbock_formula/y38yhHrk">Weitzenbock formula</a> for the Dirac Laplacian, <a href="/facts/Andr%25C3%25A9_Lichnerowicz/4AlCVKAP">André Lichnerowicz</a> deduced that if a compact spin manifold admits a metric with positive scalar curvature, its  genus must vanish. This only gives an obstruction to positive scalar curvature when the dimension is a multiple of 4, but <a href="/facts/Nigel_Hitchin/HY5u59I5">Nigel Hitchin</a> later discovered an analogous Z 2 {\displaystyle \mathbb {Z} _{2}} -valued obstruction in dimensions 1 or 2 mod 8. These results are essentially sharp. Indeed, <a href="/facts/Mikhael_Gromov_(mathematician)/N0t5hhcI">Mikhail Gromov</a>, <a href="/facts/H._Blaine_Lawson/mcTSg513">H. Blaine Lawson</a>, and Stephan Stolz later proved that the  genus and Hitchin's Z 2 {\displaystyle \mathbb {Z} _{2}} -valued analog are the only obstructions to the existence of positive-scalar-curvature metrics on simply-connected spin manifolds of dimension greater than or equal to 5. </p> <h2 id="elliptic-genus">Elliptic genus</h2> <p>A genus is called an elliptic genus if the power series Q ( z ) = z / f ( z ) {\displaystyle Q(z)=z/f(z)} satisfies the condition </p> f ′ 2 = 1 − 2 δ f 2 + ϵ f 4 {\displaystyle {f'}^{2}=1-2\delta f^{2}+\epsilon f^{4}} <p>for constants δ {\displaystyle \delta } and ϵ {\displaystyle \epsilon } . (As usual, <i>Q</i> is the characteristic power series of the genus.) </p><p>One explicit expression for <i>f</i>(<i>z</i>) is </p> f ( z ) = 1 a sn ( a z , ϵ a 2 ) {\displaystyle f(z)={\frac {1}{a}}\operatorname {sn} \left(az,{\frac {\sqrt {\epsilon }}{a^{2}}}\right)} <p>where </p> a = δ + δ 2 − ϵ {\displaystyle a={\sqrt {\delta +{\sqrt {\delta ^{2}-\epsilon }}}}} <p>and <i>sn</i> is the Jacobi elliptic function. </p><p>Examples: </p> <ul><li> δ = ϵ = 1 , f ( z ) = tanh ( z ) {\displaystyle \delta =\epsilon =1,f(z)=\tanh(z)} . This is the L-genus.</li> <li> δ = − 1 8 , ϵ = 0 , f ( z ) = 2 sinh ( 1 2 z ) {\displaystyle \delta =-{\frac {1}{8}},\epsilon =0,f(z)=2\sinh \left({\frac {1}{2}}z\right)} . This is the  genus.</li> <li> ϵ = δ 2 , f ( z ) = tanh ( δ z ) δ {\displaystyle \epsilon =\delta ^{2},f(z)={\frac {\tanh({\sqrt {\delta }}z)}{\sqrt {\delta }}}} . This is a generalization of the L-genus.</li></ul> <p>The first few values of such genera are: </p> 1 3 δ p 1 {\displaystyle {\frac {1}{3}}\delta p_{1}} 1 90 [ ( − 4 δ 2 + 18 ϵ ) p 2 + ( 7 δ 2 − 9 ϵ ) p 1 2 ] {\displaystyle {\frac {1}{90}}\left[\left(-4\delta ^{2}+18\epsilon \right)p_{2}+\left(7\delta ^{2}-9\epsilon \right)p_{1}^{2}\right]} 1 1890 [ ( 16 δ 3 + 108 δ ϵ ) p 3 + ( − 44 δ 3 + 18 δ ϵ ) p 2 p 1 + ( 31 δ 3 − 27 δ ϵ ) p 1 3 ] {\displaystyle {\frac {1}{1890}}\left[\left(16\delta ^{3}+108\delta \epsilon \right)p_{3}+\left(-44\delta ^{3}+18\delta \epsilon \right)p_{2}p_{1}+\left(31\delta ^{3}-27\delta \epsilon \right)p_{1}^{3}\right]} <p>Example (elliptic genus for quaternionic projective plane) : </p> Φ e l l ( H P 2 ) = ∫ H P 2 1 90 [ ( − 4 δ 2 + 18 ϵ ) p 2 + ( 7 δ 2 − 9 ϵ ) p 1 2 ] = ∫ H P 2 1 90 [ ( − 4 δ 2 + 18 ϵ ) ( 7 u 2 ) + ( 7 δ 2 − 9 ϵ ) ( 2 u ) 2 ] = ∫ H P 2 [ u 2 ϵ ] = ϵ ∫ H P 2 [ u 2 ] = ϵ ∗ 1 = ϵ {\displaystyle {\begin{aligned}\Phi _{ell}(HP^{2})&=\int _{HP^{2}}{\tfrac {1}{90}}{\big [}(-4\delta ^{2}+18\epsilon )p_{2}+(7\delta ^{2}-9\epsilon )p_{1}^{2}{\big ]}\\&=\int _{HP^{2}}{\tfrac {1}{90}}{\big [}(-4\delta ^{2}+18\epsilon )(7u^{2})+(7\delta ^{2}-9\epsilon )(2u)^{2}{\big ]}\\&=\int _{HP^{2}}[u^{2}\epsilon ]\\&=\epsilon \int _{HP^{2}}[u^{2}]\\&=\epsilon *1=\epsilon \end{aligned}}} <p>Example (elliptic genus for octonionic projective plane, or Cayley plane): </p> Φ e l l ( O P 2 ) = ∫ O P 2 1 113400 [ ( − 192 δ 4 + 1728 δ 2 ϵ + 1512 ϵ 2 ) p 4 + ( 208 δ 4 − 1872 δ 2 ϵ + 1512 ϵ 2 ) p 2 2 ] = ∫ O P 2 1 113400 [ ( − 192 δ 4 + 1728 δ 2 ϵ + 1512 ϵ 2 ) ( 39 u 2 ) + ( 208 δ 4 − 1872 δ 2 ϵ + 1512 ϵ 2 ) ( 6 u ) 2 ] = ∫ O P 2 [ ϵ 2 u 2 ] = ϵ 2 ∫ O P 2 [ u 2 ] = ϵ 2 ∗ 1 = ϵ 2 = Φ e l l ( H P 2 ) 2 {\displaystyle {\begin{aligned}\Phi _{ell}(OP^{2})&=\int _{OP^{2}}{\tfrac {1}{113400}}\left[(-192\delta ^{4}+1728\delta ^{2}\epsilon +1512\epsilon ^{2})p_{4}+(208\delta ^{4}-1872\delta ^{2}\epsilon +1512\epsilon ^{2})p_{2}^{2}\right]\\&=\int _{OP^{2}}{\tfrac {1}{113400}}{\big [}(-192\delta ^{4}+1728\delta ^{2}\epsilon +1512\epsilon ^{2})(39u^{2})+(208\delta ^{4}-1872\delta ^{2}\epsilon +1512\epsilon ^{2})(6u)^{2}{\big ]}\\&=\int _{OP^{2}}{\big [}\epsilon ^{2}u^{2}{\big ]}\\&=\epsilon ^{2}\int _{OP^{2}}{\big [}u^{2}{\big ]}\\&=\epsilon ^{2}*1=\epsilon ^{2}\\&=\Phi _{ell}(HP^{2})^{2}\end{aligned}}} <h2 id="witten-genus">Witten genus</h2> <p>The Witten genus is the genus associated to the characteristic power series </p> Q ( z ) = z σ L ( z ) = exp ( ∑ k ≥ 2 2 G 2 k ( τ ) z 2 k ( 2 k ) ! ) {\displaystyle Q(z)={\frac {z}{\sigma _{L}(z)}}=\exp \left(\sum _{k\geq 2}{2G_{2k}(\tau )z^{2k} \over (2k)!}\right)} <p>where σL is the <a href="/facts/Weierstrass_sigma_function/88h4u2hI">Weierstrass sigma function</a> for the lattice <i>L</i>, and <i>G</i> is a multiple of an <a href="/facts/Eisenstein_series/23vgJLFl">Eisenstein series</a>. </p><p>The Witten genus of a 4<i>k</i> dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a <a href="/facts/Modular_form/PcnbYR1f">modular form</a> of weight 2<i>k</i>, with integral Fourier coefficients. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Atiyah%25E2%2580%2593Singer_index_theorem/qwcSxamI">Atiyah–Singer index theorem</a></li> <li><a href="/facts/List_of_cohomology_theories/MkqxXM9M">List of cohomology theories</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Friedrich Hirzebruch<i> Topological Methods in Algebraic Geometry</i> <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-58663-6 Text of the original German version: <a href="http://hirzebruch.mpim-bonn.mpg.de/120/6/NeueTopologischeMethoden_2.Aufl.pdf">http://hirzebruch.mpim-bonn.mpg.de/120/6/NeueTopologischeMethoden_2.Aufl.pdf</a></li> <li>Friedrich Hirzebruch, Thomas Berger, Rainer Jung <i>Manifolds and Modular Forms</i> <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-528-06414-5</li> <li>Milnor, Stasheff, <i>Characteristic classes</i>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-08122-0</li> <li>A.F. Kharshiladze (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Pontryagin_class">"Pontryagin class"</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> <li><a href="https://www.encyclopediaofmath.org/index.php?title=Elliptic_genera">"Elliptic genera"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>McTague, Carl (2014) "Computing Hirzebruch L-Polynomials". <a href="http://www.mctague.org/carl/blog/2014/01/05/computing-L-polynomials/" target="_blank">http://www.mctague.org/carl/blog/2014/01/05/computing-L-polynomials/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Huybrechts, Daniel. "14.1 Existence, uniqueness, and embeddings of lattices". Lectures on K3 Surfaces (PDF). p. 285. <a href="https://www.math.uni-bonn.de/people/huybrech/K3Global.pdf" target="_blank">https://www.math.uni-bonn.de/people/huybrech/K3Global.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>