Topological library. Part 1: Cobordisms and their applications. Translated by V. O. Manturov.

*(English)*Zbl 1121.55003
Series on Knots and Everything 39. Hackensack, NJ: World Scientific (ISBN 978-981-270-559-4/hbk). xiv, 369 p. (2007).

This is the first of a sequence of three books collecting together and reprinting some of the most important papers in topology published in the 1950’s and 1960’s. This volume concentrates on the connections between homotopy theory and the theory of manifolds.

The book will appeal to two distinct audiences: young researchers interested in the origins and motivating questions behind the formal and concentrate algebraic topology courses taught at most graduate schools, and senior researchers already familiar with the material who will be interested in the editors’ comments and opinions.

The papers reprinted in this volume are:

L. S. Pontrjagin, “Smooth Manifolds and their Applications in Homotopy Theory”, Tr. Mat. Inst. Steklova 45 (1955; Zbl 0064.17402).

R. Thom, “Some General Properties of Differentiable Manifolds”, Comment. Math. Helv. 28, 17–86 (1954; Zbl 0057.15502).

S. P. Novikov, “Homotopy Properties of Thom Complexes”, Mat. Sb., N. Ser. 57(99), 407–442 (1962; Zbl 0193.51801).

S. Smale, “Generalized PoincarĂ©’s Conjecture in Dimensions Greater Than Four”, Bull. Am. Math. Soc. 66, 373–375 (1960; Zbl 0099.39202).

S. Smale, “On the Structure of Manifolds”, Am. J. Math. 84, 387–399 (1962; Zbl 0109.41103).

D. Quillen, “On the Formal Group Laws of Unoriented and Complex Cobordism Theory”, Bull. Am. Math. Soc. 75, 1293–1298 (1969; Zbl 0199.26705).

V. M. Bukhshtaber, A. S. Mishchenko and S. P. Novikov, “Formal Groups and Their Role in Algebraic Topology Approach”, Usp. Mat. Nauk 26, No. 2 (158), 131–154 (1971; Zbl 0224.57006).

V. M. Bukhstaber and S. P. Novikov, “Formal Groups, Power Systems and Adams Operations”, Mat. Sb., N. Ser. 84(126), 81–118 (1971; Zbl 0222.55008).

This collection, together with a translation of John Milnor’s Lectures on the \(h\)-Cobordism Theorem, was originally published in Russian in 2005. The current volume has new English translations of the four papers originally in Russian and of the one in French.

The most interesting comments included are those of M. M. Postnikov, from the 1958 Russian translation of paper # 2, and of S. P. Novikov on his paper # 3. The original reviews of the individual articles give accurate summaries of their contents.

What should be noted from the present perspective is the significance of these particular papers (apart from the fact that 4 of the 7 authors are Fields medalists). They form the basis for computations using the Adams-Novikov spectral sequences based on complex cobordism and Brown-Peterson homology and so lead the way to our current understanding of the stable homotopy groups of spheres.

The book will appeal to two distinct audiences: young researchers interested in the origins and motivating questions behind the formal and concentrate algebraic topology courses taught at most graduate schools, and senior researchers already familiar with the material who will be interested in the editors’ comments and opinions.

The papers reprinted in this volume are:

L. S. Pontrjagin, “Smooth Manifolds and their Applications in Homotopy Theory”, Tr. Mat. Inst. Steklova 45 (1955; Zbl 0064.17402).

R. Thom, “Some General Properties of Differentiable Manifolds”, Comment. Math. Helv. 28, 17–86 (1954; Zbl 0057.15502).

S. P. Novikov, “Homotopy Properties of Thom Complexes”, Mat. Sb., N. Ser. 57(99), 407–442 (1962; Zbl 0193.51801).

S. Smale, “Generalized PoincarĂ©’s Conjecture in Dimensions Greater Than Four”, Bull. Am. Math. Soc. 66, 373–375 (1960; Zbl 0099.39202).

S. Smale, “On the Structure of Manifolds”, Am. J. Math. 84, 387–399 (1962; Zbl 0109.41103).

D. Quillen, “On the Formal Group Laws of Unoriented and Complex Cobordism Theory”, Bull. Am. Math. Soc. 75, 1293–1298 (1969; Zbl 0199.26705).

V. M. Bukhshtaber, A. S. Mishchenko and S. P. Novikov, “Formal Groups and Their Role in Algebraic Topology Approach”, Usp. Mat. Nauk 26, No. 2 (158), 131–154 (1971; Zbl 0224.57006).

V. M. Bukhstaber and S. P. Novikov, “Formal Groups, Power Systems and Adams Operations”, Mat. Sb., N. Ser. 84(126), 81–118 (1971; Zbl 0222.55008).

This collection, together with a translation of John Milnor’s Lectures on the \(h\)-Cobordism Theorem, was originally published in Russian in 2005. The current volume has new English translations of the four papers originally in Russian and of the one in French.

The most interesting comments included are those of M. M. Postnikov, from the 1958 Russian translation of paper # 2, and of S. P. Novikov on his paper # 3. The original reviews of the individual articles give accurate summaries of their contents.

What should be noted from the present perspective is the significance of these particular papers (apart from the fact that 4 of the 7 authors are Fields medalists). They form the basis for computations using the Adams-Novikov spectral sequences based on complex cobordism and Brown-Peterson homology and so lead the way to our current understanding of the stable homotopy groups of spheres.

Reviewer: Keith Johnson (Halifax)