Fifty years of homotopy theory

The subject of homotopy theory may be said to have begun in 1930 with the discovery of the Hopf map. Since I began to work under Norman Steenrod as a graduate student at Chicago in 1939 and received my Ph.D. in 1941, I have been active in the field for all but the first ten years of its existence. Thus the present account of the development of the subject is based, to a large extent, on my own recollections. I have divided my discussion into two parts, the first covering the period from 1930 to about 1960 and the second from 1960 to the present. Each part is accompanied by a diagram showing the connections among the results discussed, and one reason for the twofold division is the complication of the diagram that would result were we to attempt to merge the two eras into one. The dating given in this paper reflects, not the publication dates of the papers involved, but, as nearly as I can determine them, the actual dates of discovery. In many cases, this is based on my own memory; this failing, I have used the date of the earliest announcement in print of the result (for example, as the abstract of a paper presented to the American Mathematical Society or as a note in the Comptes Rendus or the Proceedings of the National Academy). Failing these, I have used the date of submission of the paper, whenever available. Only in the last resort have I used the actual publication date. I wish to thank my many friends who have made pertinent comments, and helped refresh my memory on a number of points. Particular thanks are due to Saunders Mac Lane, William S. Massey, and Franklin P. Peterson. I also wish to acknowledge that my exposition of the solution of the immersion conjecture was based on a seminar talk by Professor Peterson on the same subject.

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[3]  A. L. Blakers,et al.  The Homotopy Groups of a Triad. III , 1951 .

[4]  J. Milnor.  On the Cobordism Ring Ω ∗ and a Complex Analogue, Part I , 1960 .

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[19]  H. Hopf.  Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension , 1935 .

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[27]  N. Steenrod.  Reduced Powers of Cohomology Classes , 1952 .

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[32]  E. Dyer,et al.  Homology of Iterated Loop Spaces , 1962 .

[33]  Larry Smith.  ON REALIZING COMPLEX BORDISM MODULES II Applications to the Stable Homotopy Groups of Spheres. , 1971 .

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[35]  I. James.  REDUCED PRODUCT SPACES , 1955 .

[36]  M. Mahowald.  A new infinite family in 2π∗S☆ , 1977 .

[37]  H. Toda.  Composition Methods in Homotopy Groups of Spheres , 1962 .

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[47]  Jean-Pierre Serre,et al.  Homologie Singuliere Des Espaces Fibres , 1951 .

[48]  Edgar H. Brown,et al.  FINITE COMPUTABILITY OF POSTNIKOV COMPLEXES , 1957 .

[49]  D. Quillen.  On the formal group laws of unoriented and complex cobordism theory , 1969 .

[50]  Duality in homotopy theory , 1955 .

[51]  K. Borsuk.  On the Lusternik-Schnirelmann category in the theory of shape , 1978 .

[52]  I. James.  ON THE SUSPENSION TRIAD , 1956 .

[53]  H. Toda,et al.  Non-triviality of an element in the stable homotopy groups of spheres , 1975 .

[54]  Hlawka.  The calculus of variations in the large , 1939 .

[55]  E. Brown,et al.  On immersions of n-Manifolds , 1977 .

[56]  J. Whitehead.  Simplicial Spaces, Nuclei and m‐Groups , 1939 .

[57]  S. Maclane,et al.  Relations Between Homology and Homotopy Groups of Spaces. II , 1945 .

[58]  J. M. Boardman,et al.  Homotopy Invariant Algebraic Structures on Topological Spaces , 1973 .

[59]  H. Toda.  On spectra realizing exterior parts of the steenrod algebra , 1971 .

[60]  G. Whitehead.  The (n + 2) nd Homotopy Group of the n-Sphere , 1950 .

[61]  H. Hopf,et al.  Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche , 1931 .

[62]  C. Wall.  Determination of the Cobordism Ring , 1960 .

[63]  H. Whitney.  The maps of an $n$-complex into an $n$-sphere , 1937 .

[64]  J. Adem.  The Relations on Steenrod Powers of Cohomology Classes , 1957 .

[65]  E. Spanier.  THE THEORY OF CARRIERS AND S-THEORY , 1962 .

[66]  S. Gitler,et al.  A spectrum whose cohomology is a certain cyclic module over the steenrod algebra , 1973 .

[67]  Jean-Pierre Serre,et al.  Groupes d'homotopie et classes des groupes abeliens , 1953 .

[68]  R. Bott.  The Stable Homotopy of the Classical Groups , 1959 .

[69]  D. Ravenel,et al.  Periodic phenomena in the Adams-Novikov spectral sequence , 1977 .

[70]  W. Massey,et al.  The 2 cohomology structure of certain fibre spaces , 1967 .

[71]  H. Whitney.  Topological properties of differentiable manifolds , 1937 .

[72]  Larry Smith.  On Realizing Complex Bordism Modules: Applications to the Stable Homotopy of Spheres , 1970 .

[73]  R. James Milgram.  Iterated loop spaces , 1966 .

[74]  J. P. May,et al.  The geometry of iterated loop spaces , 1972 .

[75]  I. James.  THE SUSPENSION TRIAD OF A SPHERE , 1956 .

[76]  J. Adams,et al.  On the Non-Existence of Elements of Hopf Invariant One , 1960 .

[77]  G. Whitehead.  Generalized homology theories , 1962 .

[78]  H. Whitney.  On the Theory of Sphere-Bundles. , 1940, Proceedings of the National Academy of Sciences of the United States of America.

[79]  N. Steenrod,et al.  Foundations of Algebraic Topology , 1952 .

[80]  E. Brown,et al.  The structure of the Spin cobordism ring , 1967 .

[81]  J. Milnor.  THE STEENROD ALGEBRA AND ITS DUAL1 , 1958 .

[82]  J. H. C. Whitehead,et al.  Duality in Relative Homotopy Theory , 1958 .

[83]  E. Brown,et al.  A universal space for normal bundles ofn-manifolds , 1979 .

[84]  W Hurewicz,et al.  ON THE CONCEPT OF FIBER SPACE. , 1955, Proceedings of the National Academy of Sciences of the United States of America.

[85]  N. Steenrod,et al.  Products of Cocycles and Extensions of Mappings , 1947 .

[86]  W. Browder,et al.  Homology operations and loop spaces , 1960 .

[87]  S. Maclane,et al.  Relations between Homology and Homotopy Groups. , 1943, Proceedings of the National Academy of Sciences of the United States of America.

[88]  W. Massey,et al.  The Homotopy Groups of a Triad. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[89]  N. Steenrod.  Topology of Fibre Bundles , 1951 .

[90]  Nobuyuki Oda,et al.  DETERMINATION OF 2-COMPONENTS OF THE 23 AND 24-STEMS IN HOMOTOPY GROUPS OF SPHERES , 1975 .

[91]  E. Brown,et al.  Relations among characteristic classes—I , 1964 .

[92]  R. Zahler,et al.  Nontriviality of the stable homotopy element °1 , 1974 .

[93]  J. Whitehead,et al.  On the Realizability of Homotopy Groups , 1949 .