is a Japanese

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, who specializes in

stable A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...

and unstable

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...

. He started publishing in 1952. Many of his early papers are concerned with the study of

Whitehead product In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in . The relevant MSC code is: 55Q15, Whitehead products and generalizations. Definition ...

s and their behaviour under

suspension Suspension or suspended may refer to: Science and engineering * Suspension (topology), in mathematics * Suspension (dynamical systems), in mathematics * Suspension of a ring, in mathematics * Suspension (chemistry), small solid particles suspend ...

and more generally with the (unstable)

homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure ...

. In a 1957 paper he showed the first non-existence result for the

Hopf invariant In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. __TOC__ Motivation In 1931 Heinz Hopf used Clifford parallels to construct the ''Hopf map'' :\eta\colon S^3 \to S^ ...

1 problem. This period of his work culminated in his book ''Composition methods in homotopy groups of spheres'' (1962). Here he uses as important tools the

Toda bracket In mathematics, the Toda bracket is an operation on homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda, who defined them and used them to compute homotopy groups of spheres in . Definition See or for ...

(which he calls the ''toric construction'') and the

Toda fibration In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dim ...

, among others, to compute the first 20 nontrivial homotopy groups for each sphere. Among his most important contributions to stable homotopy theory is his work on the existence and non-existence of so-called Toda–Smith complexes. These are finite complexes which can be characterized as having a particularly simple ordinary homology (as modules over the

Steenrod algebra In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology. For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, c ...

) or, alternatively, by having a particularly simple BP-homology. They can be used to construct the Greek letter infinite families in the stable homotopy groups of spheres. In his paper ''On spectra realizing exterior parts of the Steenrod algebra'' (1971), Toda deduced several existence and non-existence results on these complexes. The existence parts are still unsurpassed. Toda did also important work on the algebraic topology of (exceptional)

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

References

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External links

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List of Publications

Talk on Toda's Work by Frank Adams
20th-century Japanese mathematicians 21st-century Japanese mathematicians 1928 births Living people Topologists {{Japan-academic-bio-stub