On the Concept of Genus in and Complex Analysis Friedrich E. P. Hirzebruch and Matthias Kreck

he English word “genus” hails from , where it is used to connote a grouping of having com- mon characteristics. In the word is also used to objects with commonT characteristics. The concept of genus arises in various mathematical contexts, such as number theory, as well as in the we consider Figure 1. in article, topology and complex analysis. Even within the latter two areas there are various no- tions of genus that historically originated with the genus of an oriented . We begin with these (meaning compact without boundary), oriented 2 origins and afterward treat generalizations and surface F is obtained from the 2- S by modifications. We provide no detailed definitions taking repeated connected sums with the , 1 1 and proofs; rather, our goal is to give the reader T = S × S . The number of tori added is a topo- an intuitive feeling for the concept of genera. logical invariant called the genus of F, g(F), which equals Clebsch’s p. The Genus of a Surface Riemann’s concept of surface is not easy to In his paper “Theorie der Abel’schen Functionen” summarize. It is something like a ramified cover- [20] Riemann studied the topology of surfaces. He ing of the plane, and he implicitly assumes that classified a surface by looking for simple closed a surface has a sort of differentiable structure, along which to cut in to obtain a which is a great technical help. But the genus is simple presentation of the surface. He called the actually a topological invariant. One can prove this minimal number of such curves 2p and showed by using either the fundamental group or the first 2 that this invariant determines the surface. A few group H1(F). Specifically, H1(S ) = 0, years later, when Clebsch studied surfaces from and the formula a more algebraic geometric viewpoint, he called p H (F♯T ) ≅ H (F) ⊕ Z2, “das Geschlecht (genus)” of the surface. 1 1 In more modern terms one can formulate Rie- where ♯ stands for connected sum, implies that 2 mann’s insight as follows: every connected, closed a surface Fn obtained from S by connected sum with n tori has Friedrich E. P. Hirzebruch is professor emeritus at the 2n Max-Planck-Institut für Mathematik, Bonn, Germany. His H1(Fn) ≅ Z . email address is [email protected]. The rank of the k-th homology group is called the Matthias Kreck is a professor at the Hausdorff Research k-th Betti number, Institute for Mathematics at the University of Bonn. His email address is [email protected]. bk(X) := rank(Hk(X)),

June/July 2009 Notices of the AMS 713 and so we obtain the formula with countable basis has a triangulation, and then a proof in the combinatorial world is not difficult. g(F) = b1(F)/2. Instead of the Betti number one can use the What Is the Analytic Significance of the , Genus? i e(X) := X(−1) bi(X), The main intention of Riemann’s topological i considerations was to study surfaces as objects to determine the genus of F, namely, b0(F) = in complex analysis: in modern terms, as complex b2(F) = 1, and so e(F) = 2 − b1(F) = 2 − 2g(F), of complex dimension 1, complex implying curves. A complex is a topological g(F) = 1 − e(F)/2. manifold (meaning a topological Hausdorff space The Euler number itself can be computed with countable basis locally homeomorphic to Rn) combinatorially without referring to homology. together with an atlas whose coordinate changes Every surface has a triangulation (see next sec- are holomorphic maps. We note that in complex tion), which we can visualize by putting a net of dimension 1 a countable basis follows from the triangles over F existence of a complex structure [19]. On such a complex Riemann considered divisors D, which are finite formal linear combinations of points in the surface with coefficients in Z. Each meromorphic function on a closed surface (in the following we assume that the surfaces are closed) determines a divisor D by its zeroes and poles (counted with multiplicity). To each divisor we attach the sum of the coefficients called deg(D). Given D we can consider the vector space of meromorphic functions characterized by the property that its divisor plus the given divisor does not have negative multiplicities. This vector space is finite-dimensional, and its dimension is Figure 2. denoted by l(D). For a divisor D Riemann [20] proved his in- equality and then l(D) ≥ deg(D) + 1 − g.

e(F) = number of vertices − number of edges This is the starting point of the famous Riemann- Roch theorem [21], which gives an equality using + number of triangles. the canonical divisor K: However, to show that the combinatorially defined l(D) − l(K − D) = deg(D) + 1 − g. Euler characteristic is a topological invariant needs a proof. Why is this an important result? If we look at the right-hand side of the equation, we see the sum What Is the Topological Significance of the of a very simple invariant of a divisor, its degree, Genus? and a topological invariant, 1 − g. However, the So far we have attached a topological invariant, a left-hand side is a complicated analytic invariant: certain characterization, to a surface: the genus. the difference of dimensions of certain function How powerful is this invariant? What is its topo- spaces, more precisely spaces of meromorphic logical significance? One can interpret Riemann as functions with restricted zeroes and poles. saying that a differentiable surface is determined by its genus. A modern proof of this can be ob- The of Algebraic tained from elementary Morse theory [6], [7]. But a Varieties stronger statement is true: the genus characterizes It should be noted that the passage from dif- the homeomorphism . This result was proved ferentiable structures to complex structures is a by Rado long after Riemann’s work: dramatic change, since there are in general many Theorem [19]. Two connected closed oriented sur- different complex structures on a given surface. faces F and F ′ are homeomorphic if and only if One has a of complex structures ′ which for g ≥ 2 is itself a complex manifold of g(F) = g(F ). complex dimension 3g − 3. For g = 0 the moduli This is not an easy result. The central step in space is a point, and for g = 1 it has complex Rado’s proof is to show that a topological surface dimension 1.

714 Notices of the AMS Volume 56, Number 6 One can proceed further and impose even more The Todd Genus refined structures on a surface, for example, com- The fourth definition of the arithmetic genus plex algebraic structures. Whereas topological and was given by J. A. Todd [24]. A canonical divi- smooth manifolds were intensively studied in the sor of a smooth projective algebraic of first half of the last century, complex manifolds dimension n is given as a divisor of a meromor- of dimension greater than 1 were not investigated phic n-. It is an algebraic cycle of topological very much by complex analytic methods (of course codimension 2. Todd introduced geometric canon- the function theory of several complex variables ical cycles for all even codimensions. He defined in open domains in Cn was a subject of great polynomials in these cycles, where the product is interest). In contrast, algebraic varieties, the set of given by intersections. The n-th Todd polynomial zeroes of a of polynomials, were the subject is of codimension 2n and represents for a variety of constant mathematical investigations, at least of dimension n a certain number called the Todd by constructing interesting examples and studying genus. Todd believed that his genus was the same their geometry. Many formulae were found that as the arithmetic genus, but rigorous justification of this fact came much later. led to interesting questions and conjectures. The Todd canonical classes represent homol- In this context another genus, the arithmetic ogy classes, and they are up to signs Poincaré genus, played an important role. In the early dual to the Chern classes of the tangent bundle of 1950s four definitions of the arithmetic genus of the variety [18]. More generally, Chern classes are a projective smooth algebraic variety V of com- defined for complex vector bundles. In contrast to plex dimension n were known. The first two are complex manifolds, where holomorphic maps play denoted by pa(V) and Pa(V). Severi conjectured a definitive role, complex vector bundles are pure- that these numbers agree and can be computed in ly topological objects; roughly speaking, they are terms of the dimension gi (V) of the vector space a family of complex k-dimensional vector spaces of holomorphic differential forms of degree i: parametrized by the points of a topological space X. There is also a topology on the disjoint union of pa(V) = Pa(V) = gn(V) − gn−1(V) these vector spaces, and the key property is that n−1 locally this family of vector spaces is homeomor- + · · · + (−1) g1(V). phic to a product U ×Ck, where U is an open subset The expression on the right-hand side is the third of X. The formal definition of Chern classes is too definition, which we recommend to the reader complicated for an article such as this one, but (actually in a slightly modified form described the basic idea behind them can be explained by below). Using sheaf theory, Kodaira and Spencer considering the case of a differentiable complex [14] proved that the three expressions agree. vector bundle E over a closed differentiable man- 2i The expression on the right looks like an Euler ifold X. Then the Chern ci(E) ∈ H (X; Z) is characteristic, but in a strange form. The “correct” the first obstruction to the existence of k − i + 1 Euler number is the holomorphic Euler number, linearly independent sections on E. If one chooses, n for example, a single on E and the choice is i χ(V) := X(−1) gi (V), generic, then the set of zeroes is a submanifold of i=0 X of dimension dimX − 2 dimE. Thus we obtain a homology class whose Poincaré dual sits in H2k(X) called the arithmetic genus. The number of com- and is the k-th Chern class c (E) of E. If X is a ponents of V is g (V). Thus for a connected variety k 0 closed complex k-dimensional manifold and E its 1 + (−1)np (V) = χ(V). Often g (V) is called the a n complex tangent bundle, then c (E) evaluated on geometric genus of V. For the case of a curve k the fundamental cycle is the Euler characteristic (Riemannian surface) we have of X by the Poincaré-Hopf theorem. The Todd genus in terms of the Chern classes of g1(V) = g(V), the tangent bundle is the evaluation of a certain ra- and so the geometric genus and Riemann’s genus tional polynomial in the Chern classes T (c1, c2,...) agree. on the fundamental class. To motivate the con- Both the arithmetic and geometric genus are struction of these polynomials (which are rather multiplicative: complicated expressions), we note that if the Todd genus agrees with the arithmetic genus, then cer- × ′ = ′ χ(V V ) χ(V)χ(V ) tainly for complex projective spaces (where the and arithmetic genus takes the value 1) they have to agree. Furthermore, the Todd genus has to be g (V × V ′) = g (V)g (V ′), n+m n m multiplicative in a way that reflects the multiplica- where n = dim V and m = dim V ′. tivity of the arithmetic genus, and so it should be The gi (V) are birational invariants [25], and so a multiplicative sequence in the sense of [8]. This the arithmetic genus is a birational invariant. was the motivation for the first author to introduce

June/July 2009 Notices of the AMS 715 the general concept of multiplicative sequences of These properties of the Todd genus motivated polynomials. He characterizes the Todd sequence the first author to introduce the general concept by a special multiplicative sequence with value 1 of genus. This is an invariant for certain classes on each complex projective space. of manifolds in terms of characteristicΦ classes The first four polynomials are of the tangent bundle (perhaps equipped with a 1 stable almost complex structure) with values in a T := c , 1 2 1 ring fulfilling the two properties above. We note thatΛ for a Riemannian surface F the Todd genus 1 2 T2 = (c + c2), is c (F)/2, which is half the Euler characteristic 12 1 1 1 of F, namely, 1 − g(F). Thus the Todd genus is T3 = c1c2, in this case essentially the genus of a Riemannian 24 surface. 1 2 2 4 T4 = (−c4 + c3c1 + 3c + 4c2c − c ). An important invariant of oriented manifolds 720 2 1 1 is the signature (b+ − b−), which is the signature in Theorem [8]. Let V be a nonsingular compact com- the sense of linear algebra of the intersection form plex algebraic variety of dimension n. Then of a 4k-dimensional closed oriented manifold M (if χ(V) = hTn(V), [V]i, the dimension is not divisible by 4, the signature is defined as zero). It is denoted by the evaluation of the Todd polynomial Tn on the fundamental class. sign(M) ∈ Z. We see that this result has a similar flavor to the The first author was looking for a formula Riemann-Roch Theorem, since it relates analytic that, in analogy to the formula for the arithmetic information, the holomorphic Euler number, to a genus, computes the signature in terms of char- topologically defined invariant, the Todd genus. acteristic classes. This was done at a time when If the complex dimension of V is 1, the case the Riemann-Roch formula was only conjectured. of a Riemannian surface, the theorem above is a In fact, the signature theorem became an impor- special case of the Riemann-Roch formula above, tant ingredient in the proof of the Riemann-Roch namely, the case where D = 0. In the same sense, formula. Since there is no complex structure on the theorem above is the special case of the the tangent bundle, one has to use the Pontrjagin Hirzebruch-Riemann-Roch formula for D = 0 [8]. 4i classes (M) ∈ H (M) instead of the Chern class- In 1957 Grothendieck generalized this by consid- es. These are (up to sign) the Chern classes of the ering a parametrized version of the Riemann-Roch complexification of the tangent bundle. In analogy formula [2]. All this is a long story, and although to the Todd genus, the first author used his for- closely related to genera, it would lead us too far malism of multiplicative sequences to construct away from our main themes. polynomials in Pontrjagin classes that take for each even-dimensional complex projective space Bordism and Generalized Genera the value 1, the signature of M. These are the When we said that the Todd genus is topologically L-polynomials. The first three L-polynomials are defined, this was too brief. In addition to the 1 underlying differentiable manifold, one needs a L1 = p1, complex structure in a weaker sense: a complex 3 structure on the sum of the tangent bundle with a 1 2 L2 = (7p2 − p ), trivial bundle. This structure is called a stable al- 45 1 most complex structure. A manifold with a stable 1 L = (62p − 13p p + 2p3). almost complex structure is called a stable almost 3 945 3 2 1 1 complex manifold. Note that with this definition, If, as for the arithmetic genus, one knew that the even an odd-dimensional manifold can have a values on the even-dimensional projective spaces stable almost complex structure. characterize the signature, one would obtain the Since the Chern classes are stable invariants, desired formula. This would follow if, after pass- which means that they are unchanged if we add a ing to a multiple if necessary, each manifold were trivial (complex) bundle, the structure one needs bordant to a linear combination of products of for defining the Todd genus is a stable almost projective spaces. The reason is that both the complex structure. For such manifolds the Todd signature and the L-polynomials are bordism in- genus has the following fundamental properties: variants. Here two oriented manifolds M and N are • it is additive (i.e., the Todd genus of a bordant if there is a compact oriented manifold W disjoint union is the sum of the Todd with boundary being the disjoint union of M and genera) −N, the manifold N with the opposite orientation. • it is multiplicative (i.e., the Todd genus The bordism classes of closed oriented of a product is the product of the Todd n-dimensional manifolds form a group un- genera). der disjoint union, denoted by n. The sum Ω

716 Notices of the AMS Volume 56, Number 6 ∗ := Pn n is a ring with respect to the many simply connected 4-manifolds M that have productΩ ofΩ two manifolds. Thom [23] computed an exotic smooth structure, which means there ∗ ⊗ Q. It is the polynomial ring with generators, exists another manifold homeomorphic but not theΩ even-dimensional projective spaces. These diffeomorphic to M. The first example was found considerations lead to: by Donaldson [4]. Later on, his techniques were Theorem (Signature Theorem) [8]. Let M be a applied to show that many simply connected closed smooth oriented manifold. Then 4-manifolds have infinitely many smooth struc- tures, for example, the K3-surface {x ∈ CP3| x4 = sign(M) = hL(M), [M]i. P i 0} (a complex surface, so the real dimension is 4). Returning to the Todd genus, we noted that The following is one of the big open prob- it is defined for closed manifolds with stable al- lems in differential topology: Is there any closed most complex structure. In analogy to the bordism 4-manifold with no exotic structure? The most groups of oriented manifolds, Milnor [16] defined interesting examples would be the complex pro- and computed bordism groups of stable almost jective plane CP2 or the 4-sphere S4. If S4 has complex manifolds. The answer is simpler than a unique smooth structure, this is the smooth for oriented bordism groups: the bordism ring of 4-dimensional Poincaré conjecture, which one can stable almost complex manifolds is a polynomial formulate as follows: A closed smooth simply con- ring over Z in variables xi corresponding to stable nected 4-manifold with Euler characteristic 2 is almost complex manifolds of real dimension 2i, diffeomorphic to S4 (it has automatically second which Milnor explicitly describes. If one takes the Betti number 0 and so signature 0; thus by the tensor product with Q, generators are given by theorem above it is homeomorphic to S4). the projective spaces CPi . The existence of infinitely many smooth struc- tures on a closed manifold is something that The Relevance of the Signature happens exclusively in dimension 4. In all oth- Here we can only indicate some aspects (in a er dimensions this number is finite. This result nonhistorical order). The classical genus of a is closely related to the Hauptvermutung, which Riemannian surface completely characterizes the says that if a topological manifold can be trian- homeomorphism type (which for surfaces agrees gulated, then this triangulation is unique up to with the diffeomorphism type). In a certain sense refinement. This is not true (the first counterexam- one has an analogous result for closed smooth ples were given by Milnor [17]), but in dimension simply connected 4-manifolds. > 4 the work of Kirby and Siebenmann about Theorem [5], [3]. Two closed differentiable simply the Hauptvermutung [13] shows that a topological connected 4-manifolds are homeomorphic if and manifold of dimension > 4 has at most finitely only if the Euler characteristic, the signature, and many piecewise linear structures. Using surgery the type (even or odd) agree. theory, one can show that a piecewise linear man- Here the type is even if and only if all self- ifold of dimension > 4 has at most finitely many intersection numbers are even. This is a very deep smooth structures. Combining these two results, result based on independent difficult theorems one sees that a topological manifold of dimension by Freedman and Donaldson. Freedman classified > 4 has at most finitely many smooth structures. simply connected topological 4-manifolds in In this last result the classification of smooth terms of the intersection form and a Z/2-valued structures on plays an essential role (for invariant, the Kirby-Siebenmann invariant. This the first examplesbyMilnor, see [15]; for the gener- vanishes for smooth manifolds as well as for al classification in dimension > 4 by Kervaire and manifolds homotopy equivalent to S4. Thus, as Milnor, see [12]). The signature and in particular a special case, Freedman proves the topological the signature theorem are as much central tools 4-dimensional Poincaré conjecture: A 4-dimensional for the existence of exotic structures on spheres manifold homotopy equivalent to S4 is home- as tools for the classification of such structures. omorphic to S4. Every unimodular symmetric We indicate this for existence. Milnor constructs bilinear form is the intersection form of a closed certain compact smooth manifolds W with bound- − simply connected topological 4-manifold, and the ary homeomorphic to S4n 1. Then he considers classification of such forms is unknown. But for the union of W with the cone over its boundary. If smooth manifolds Donaldson used gauge theory the boundary is diffeomorphic to S4n−1, then this to show that the intersection forms are very is a smooth manifold, and thus one can compute special and, because of some classical results, its signature by the signature theorem in terms are classified by the rank (equivalent to the Euler of the L-polynomials. Except for the expression characteristic), the signature, and the type. in the top Pontrjagin class pn, all other terms in In contrast to Riemann surfaces, the analogous the L-polynomial can be computed in terms of result for a diffeomorphism classification is the Pontrjagin classes of W . Thus one can use completely different in dimension 4. There are the signature formula to compute the term in pn

June/July 2009 Notices of the AMS 717 in the L-polynomial. The coefficient of pn in the form has determinant ±1) of the E8-form, we find L-polynomial is a rational number, that the Mayer-Vietoris sequence implies that the 2n 2n−1 boundary of W (E8) is a homotopy sphere. Appli- 2 (2 − 1) n−1 (−1) b2n, cation of the Poincaré conjecture proved by Smale 2n! [22] shows that the boundary is homeomorphic where b is the Bernoulli number. If the boundary 2n to S11. We now apply the signature theorem to of W is diffeomorphic to S4n−1, the fact that p n show that it is not diffeomorphic to S11 follow- is an gives a certain congruence between ing the principle explained above. If there were a the difference of the signature of the union of W diffeomorphism f : ∂W (E ) → S11, then we could with the cone over the boundary and the other 8 consider M := W (E ) ∪ D12 and obtain a smooth expressions in the L-polynomial (we will carry 8 f manifold whose homology is trivial except in de- out an especially simple example). If this congru- gree 0, 6, and 12. The intersection form of this ence does not hold, the boundary of W is not manifold is by construction the E -form, whose diffeomorphic to the sphere. This way one can 8 signature is 8. Now we apply the signature theorem produce examples of exotic structures on spheres and obtain of dimension 4n − 1 for n > 1 (see also [17]). We want to use plumbing [9] similar to a 62 8 = hp3(M), [M]i, construction used by Milnor to give explicit ex- 945 amples of exotic spheres and at the same time a contradiction (note that the only potentially the construction of a topological manifold without nontrivial Pontrjagin class is p3(M), an integral co- smooth structure (the existence of such manifolds homology class). Thus ∂W (E8) is an exotic sphere. was first shown by Kervaire [11]). We consider the If instead of a diffeomorphism we use a homeo- E8-graph morphism, we obtain a topological manifold M, which by the same argument as above cannot admit a smooth structure. The use of the E8-graph is motivated by the fact that the resolution of the singularity in (0, 0, 0) of 2 3 5 z1 +z2 +z3 = 0 consists of eight nonsingular ratio- Using E8, we construct a manifold with bound- nal curves of self-intersection number −2 whose ary of dimension 12 by gluing together for each intersection behavior is given by E8. edge a copy of the disc bundle of the tangent bun- 6 dle of S using the following recipe. If two vertices The Atiyah-Singer Index Theorem and are joined by an edge, we take a trivialization of Other Genera the disc bundle over a disc D6 in S6 to obtain an embedding of D6 × D6 into the disc bundle, where Except for the signature, the left-hand sides of our the first component maps to S6 and the second formulas related to the genus were of an analytic to the fibres. Then we identify (x, y) in the first nature, being given by dimensions of certain vec- product with (y, x) in the second: spaces of functions or differential forms. In fact the signature can also be interpreted as an analytic invariant via Hodge theory. It is the index of a differential operator, the signature operator. Thus the signature theorem is an index theorem expressing the index of an elliptic differential operator in topological terms. During the 1960s Atiyah and Singer [1] proved a general index theorem for elliptic differential operators on smooth manifolds extending for example the signature theorem. Besides the signa- ture the most important operators are the Laplace operator whose index is the Euler characteris- tic and the Dirac operator on a manifold with spin-structure. The topological side of the index Figure 3. formula for the Dirac operator, the Aˆ-genus, was studied in [8]. In the 1980s Ochanine and Witten defined very interesting genera for spin, respec- The result is a compact 12-dimensional manifold tively string, manifolds, the Ochanine, respectively W (E8) with boundary and corners, which one can Witten, genus. The remarkable property of these smooth. By construction it is homotopy equiva- genera is that they take values in rings of mod- lent to a wedge of eight copies of S6. By general ular forms (compare, for example, [10]). One can position the fundamental group is trivial. Using conjecture that these genera are only a shadow of the unimodularity (meaning that the intersection interesting new (co)homology theories associated

718 Notices of the AMS Volume 56, Number 6 with the term elliptic . In the end one [18] I. R. Porteous, Todd’s canonical classes, Pro- expects that elliptic cohomology will play a cen- ceedings of Liverpool Singularities Symposium I, tral role in index theory on the loop space of a (1969/70), pp. 308–312. Lecture Notes in Math., Vol. 192, Springer, Berlin, 1971. manifold in analogy with the role that K-theory [19] T. Rado, Über den Begriff der Riemannschen Fläche, plays in index theory on smooth manifolds. Acta Szeged 2 (1925), 101–121. Acknowledgment. We would like to thank the [20] B. Riemann, Theorie der Abel’schen Functionen, J. reviewer for a careful reading and very helpful Reine Angew. Math. 54 (1857), 101–155. [21] G. Roch, Ueber die Anzahl der willkürlichen comments. Constanten in algebraischen Functionen, J. Reine Angew. Math. 64 (1865), 372–376. References [22] Stephen Smale, Generalized Poincaré’s conjecture [1] Michael F. Atiyah and I. M. Singer, The index of in dimensions greater than four, Ann. of Math. (2) elliptic operators. I, Ann. of Math. (2) 87 (1968), 484– 74 (1961), 391–406. 530; The index of elliptic operators. III, Ann. Math., [23] René Thom, Quelques propriétés globales des (2) 87 (1968), 546–604. variétés différentiables, Comment. Math. Helv. 28 [2] Armand Borel and Jean-Pierre Serre, Le (1954), 17–86. théorème de Riemann-Roch, Bull. Soc. Math. [24] J. A. Todd, The arithmetical invariants of algebra- France 86 (1958), 97–136. ic loci, Proc. London Math. Soc. (2), Ser. 43, 1937, [3] S. K. Donaldson, An application of gauge theory to 190–225. four-dimensional topology, J. Differential Geom. 18 [25] Bartel L. van der Waerden, Birational invari- (1983), 279–315. ants of algebraic manifolds, Acta Salmanticentsia. [4] , Irrationality and the h-cobordism conjec- Ciencias: Sec. Mat., no. 2 (1947); Birationale Trans- ture, J. Differential Geom. 26 (1987), 141–168. formation von linearen Scharen auf algebraischen [5] Michael Hartley Freedman, The topology of Mannigfaltigkeiten, Math. Z. 51 (1948), 502–523. four-dimensional manifolds, J. Differential Geom. 17 (1982), 357–453. [6] André Gramain, Topologie des surfaces, Collection SUP: “Le Mathématicien”, 7, Presses Universitaires de France, , 1971. [7] Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York-Heidelberg, 1976. Medicine and Mathematical Humor [8] F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathe- Citations to Notices articles crop up in many matik und ihrer Grenzgebiete, 9, Springer-Verlag, places—but in a medical journal? In 2006, Best Berlin-Göttingen-Heidelberg, 1965. Practice & Research Clinical Obstetrics and Gyn- [9] , Gesammelte Abhandlungen. Band I, Zur Theorie der Mannigfaltigkeiten, Springer-Verlag, aecology carried an article called “Assessing new Berlin, 1987. interventions in women’s health”. Written by [10] Friedrich Hirzebruch, Thomas Berger, and University of Birmingham biostatistician Robert Rainer Jung, Manifolds and modular forms K. Hills and research fellow Jane Daniels, the ar- (translated by Peter S. Landweber), Aspects of ticle discusses principles for running or assessing Mathematics, E 20, Vieweg, Wiesbaden, 1992. clinical trials of medical treatments for women, [11] M. A. Kervaire, A manifold which does not admit with an emphasis on how to evaluate the treat- any differentiable structure, Comment. Math. Helv. ments’ efficacy in the face of unclear or conflict- 34 (1960), 257–270. ing trial results. “[I]t is not acceptable to rely on [12] Michel A. Kervaire and John W. Milnor, Groups proof by anecdotal evidence, eminent authority, of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. or vigorous handwaving,” the authors write, cit- [13] Robion C. Kirby and Laurence C. Sieben- ing the widely read Notices article “Foolproof: A mann, Foundational essays on topological man- sampling of mathematical folk humor”, by Alan ifolds, smoothings and triangulations, Annals of Dundes and Paul Renteln (January 2005). As he Mathematics Studies, 88, Princeton University later explained to Notices Editor Andy Magid, Press, Princeton, NJ, and University of Tokyo Press, Hills was an undergraduate mathematics student 1977. at the University of California, Los Angeles, and [14] K. Kodaira and D. C. Spencer, On arithmetic gen- heard jokes about the various methods of proof era of algebraic varieties, Proc. Nat. Acad. Sci. U.S.A. (such as “proof by intimidation”), many of which 39 (1953), 641–649. are listed in the Dundes-Renteln piece. Hills [15] John W. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399–405. and Daniels also quote Bertrand Russell: “The [16] , On the cobordism ring ∗ and a complex fact that an opinion has been widely held is no analogue. I, Amer. J. Math. 82 (1960),Ω 505–521. evidence whatever that it is not utterly absurd.” [17] , Collected papers of John Milnor. III. Dif- ferential topology, American Mathematical Society, Providence, RI, 2007.

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