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Differential Algebraic Topology: From Stratifolds to Exotic Spheres

American Mathematical Society
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A stratifold is “a differential space with certain properties” which take a bit of doing even to state (not too much, though: it’s not as bad as the singular spaces of Goresky and MacPherson, Whitney stratifications, or other such conceits one encounters in differential topology all too often): in rough terms a stratifold S is rigged so that “[it naturally decomposes] into subspaces which are smooth manifolds.” A bit more precisely, a stratifold S a disjoint union of strata Si characterized by a condition that each such Si is the set of all points of S at which the corresponding tangent space is i-dimensional (in the manifold sense: this is what these beasties are, by hypothesis — see immediately below). Each such stratum is dealt the data of a differential space by attaching a “locally detectable subalgebra of the [ambient] algebra of continuous functions” to it, permitting the requirement that each at each index i one gets a smooth manifold. For S to be k-dimensional one has to have that k ≥ i for all i (and presumably k is the maximum of all the i’s). Then one uses bump functions so as to finesse some sort of coherence vis à vis open sets in the strata vs open sets in S itself, while S is presupposed locally compact and the partial unions of the strata, called the r-skeletons (just let i run up to r to get the r-th one), are required to be closed in S.

Well, why would one want to do all this, however, even if it’s not as nasty as Whitney stratification? Says Kreck: “The way Poincaré introduced homology in [his 1895 J. École Polytechnique classic, Analysis Situs] is the model for our approach.” He goes on: “Poincaré’s original idea … came up again many years later, when in the 1950s Thom invented and computed the bordism groups of smooth manifolds. Following on Thom, Conner and Floyd introduced singular bordism … in the 1960s. This homology theory is much more complicated than ordinary homology, since the bordism groups associated to a point are complicated abelian groups, whereas [as every one knows] for ordinary homology they are trivial except in degree 0.”

It is the case, in point of fact, that Whitney stratification (as already maligned above) is the most important example of ploys used to pass from singular bordism to ordinary homology, so to have an alternative available in stratifolds, “present[ing] an approach to … homology which reflects the spirit of Poincaré’s original idea,” is a wonderful thing. And Differential Algebraic Topology: From Stratifolds to Exotic Spheres is offered, to boot, as an introductory text — with the subject of bordism occuring smoothly and early on p.45. Moreover, “[i]t is rather easy and intuitive to derive the basic properties of homology in the world of stratifolds … [w]e also define sratifold cohomology groups by following an idea of Quillen, who gave a geometric construction of cobordism groups, the cohomology theory associated to singular bordism … [and] certain important cohomology classes occur very naturally in this description, in particular the characteristic classes of smooth vector bundles over smooth oriented manifolds …” This is clearly a huge selling point for the stratifolds approach, and the following closes the deal: “Another useful aspect of this approach is that one of the most fundamental results, … Poincaré duality, is almost a triviality.”

Kreck pitches his discussion at the level of “[r]eaders … familiar with the basic notions of point set topology and of differential topology,” and this is a fair appraisal of the situation. It obviously helps a great deal if the reader likes manifolds on a personal level (I particularly like Loring Tu’s An Introduction to Manifolds; see my review in this column), and some familiarity with the staples of topology, e.g. Mayer-Vietoris, Brouwer’s fixed point theorem, Künneth, etc., would be useful, too. Additionally it would be good, also, if the words, “Euler classes, Chern classes, Stiefel-Whitney classes and Pontryagin classes,” do not cause panic; here must mention, of course, the uncontested classic by Milnor and Stasheff, Characteristic Classes, but, to be sure, these very topics are covered in the latter chapters of the book under review. Thereupon Kreck turns his attention to things even more Milnor, namely, exotic 7-spheres; the book ends with a chapter discussing connections with ordinary singular (co)homology and a trio of appendices on, respectively, constructing stratifolds, proving Mayer-Vietoris, and the tensor product. All good stuff.

And Differential Algebraic Topology: From Stratifolds to Exotic Spheres is a good book. It is clearly written, has many good examples and illustrations, and, as befits a graduate-level text, exercises. It is a wonderful addition to the literature.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Wednesday, May 19, 2010
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Matthias Kreck
Graduate Studies in Mathematics 110
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Michael Berg
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Monday, June 21, 2010