Elliptic Cohomology (University Series in Mathematics)

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Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of the book is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from `Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalisations and potential applications.

Elliptic Cohomology (University Series in Mathematics) Review

In differential topology one of the most interesting and challenging objects to study is the oriented cobordism ring, which was first defined and studied by the mathematician Rene Thom. This construction arises when considering two smooth compact oriented manifolds M and M' which belong to the same 'oriented cobordism class'. This means that there exists a smooth compact oriented with boundary whose boundary (with the induced orientation) is diffeomorphic to the disjoint union of M and -M', where -M' is the manifold M' with the opposite orientation. It is straightforward to show that oriented cobordism is an equivalence relation and that the collection of oriented cobordism classes of manifolds is an Abelian group with group operation the disjoint union. In addition, the operation of Cartesian product makes the sequence (ordered under dimension) of oriented cobordism a graded ring: the 'oriented cobordism ring.' Tools from algebraic topology, such as the theory of characteristic classes (more specifically, the Pontryagin classes), are used to study the oriented cobordism ring. Another construction used to study this ring is the universal Thom space, which is a kind of generalization of the universal vector bundle (or "classifying space") in homotopy theory. One can show that the m-dimensional homotopy groups of the universal Thom space (with a base point) are canonically isomorphic to the oriented cobordism group. This fact allows one to prove that the oriented cobordism group is finite for non-zero dimension modulo 4, and is finitely generated with rank equal to the number of partitions of r, when the dimension is equal to 4r. If the oriented cobordism ring is tensored with the rational numbers, thus removing torsion, then one can show that this tensor product is a polynomial algebra with generators CP(2), CP(4), CP(6), .... where CP(n) is complex n-dimensional projective space. The sequence CP(2), CP(4), .... is an example of what is called a 'basic sequence' of manifolds.

The characteristic classes used to distinguish one manifold from another usually involve relations between polynomials, and to establish the properties of these polynomials once and for all, the mathematician F. Hirzebruch introduced 'multiplicative sequences' of polynomials. The nth term of a multiplicative sequence is a homogeneous polynomial of degree n and they can be used to describe the relations between the characteristic classes, such as the Pontryagin, Chern, and Stiefel-Whitney classes, of vector bundles. Working with the characteristic classes of vector bundles reveals that they frequently have the form of power series. Multiplicative sequences can be uniquely associated with power series as Hirzebruch showed, and those sequences with rational coefficients are of particular interest since they are used to define the 'genus' of a compact, oriented, smooth manifold. One can show that the correspondence between the manifold and its genus gives rise to a ring homomorphism between the cobordism ring and the rational numbers Q (i.e. a homomorphism from the tensor product of the cobordism ring with Q to Q). This result gives rise to the famous Hirzebruch signature for manifolds whose dimension is a multiple of 4, which can be written in terms of a linear function of the Pontryagin numbers. The multiplicative sequence of polynomials in this case is associated to the hyperbolic tangent function with power series involving the Bernoulli numbers, and the Hirzebruch signature can be shown to be equal to the genus (the 'L-genus') arising from the multiplicative sequence, which it turns out is always an integer.

This book begins by showing that these kinds of constructions dealing as they do with hyperbolic functions (and their corresponding 'addition formula') are special cases of what that author calls the 'universal elliptic genus.' The L-genus is a "degenerate" case of the elliptic genus (with the famous "A-genus" being another degenerate case). This degeneracy is related somewhat loosely to the fact that the ordinary trigonometric functions and the hyperbolic functions are degenerate cases of elliptic functions. The genus is actually defined for integral domains R over Q, and is a ring homomorphism from the tensor product of the oriented cobordism ring with Q to R (to be called here the rationalized cobordism ring), that maps the identity element of this tensor product to the identity element of R. The author shows how every genus arises from a power series by using the logarithm associated with the genus and the notion of a 'formal group law' (which is essentially a generalization of the trigonometric and hyperbolic addition formulas). The logarithm determines the formal group law, and using this the author gives conditions, involving power series, for finding when one genus has the same logarithm as another. The L-genus falls out for a power series expansion of x/tanh(x) (with formal group law the addition formula for the hyperbolic tangent), while one obtains the A-genus for the power expansion of (x/2)/sinh(x/2) (with formal group law the addition formula for sinh(x+y)). The author then shows that these are degenerate cases of the universal elliptic genus whose addition formula arises from the problem of doubling the arc length along a lemniscate and whose logarithm depends on two parameters 'd' and 'e'. When d = -1/8, e = 0 one captures the A-genus, whereas the L-genus arises when d = e = 1. These examples motivate the definition of a 'universal formal group law', with the designation as 'universal' arising from the fact that the group law is respected by all homomorphisms from the rationalized cobordism ring to R. An important property of the elliptic genus is that it is 'multiplicative' with respect to Cartesian products and locally trivial fibrations. The genus is called 'strongly multiplicative' if it is multiplicative for spin fibrations whose structural groups are compact and connected. The author shows that if a genus is strongly multiplicative, then it is elliptic. The proof involves the use of 'Milnor manifolds', which are essentially fibrations where the total space and base space are complex projective spaces. The use of Milnor manifolds is perhaps not surprising since the elliptic genus is determined by its values on complex projective spaces (specifically on CP(2) and (quaternionic) HP(2)).

The goal of the author is to show that there are cases where the elliptic genus is 'nondegenerate'. He does this in the context of the sporadic Mathieu group after spending a few chapters elaborating on how to construct elliptic cohomology theory. Elliptic cohomology theory is an "extraordinary" cohomology theory, in that the dimension axiom does not hold. The key to understanding these early chapters are the grasp of the notions of a 'multiplicative' cohomology theory on finite groups and a generalization of character theory on (finite) groups called the 'Mackey functor.' A multiplicative cohomology theory is one where each of the cohomology groups has the structure of a graded commutative ring. Complex K-theory is a multiplicative cohomology theory where the multiplication arises from the tensor product of complex vector bundles. For a ring A and a finite group G, the Mackey functor is a functor from the category of subgroups of G (with morphisms generated by inclusions and conjugations) to the family of A-modules which has induction and restriction maps and that has a double coset formula. A map from the subgroups of G to the (complex-oriented) cohomology of their classifying space is a Mackey functor, and the author shows that in fact it is a 'Green functor' in that the family of modules has a multiplicative structure and the restriction and induction maps satisfy Frobenius reciprocity. This gives rise to a group cohomology that generalizes that of the ordinary cohomology of the classifying space BG of the group G. The author shows how to compute this cohomology for BG, the first result being that for cyclic prime subgroups of G, the (complex oriented) cohomology is generated by the Chern classes. The author then shows that if one specializes to the case of elliptic cohomology, then for odd primes less than 24, the elliptic cohomology of the classifying space of the Mathieu group with 24 elements is concentrated in even dimensions.

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