Hiroshi Toda
Encyclopedia
is a Japan
ese mathematician
, who specializes in stable
and unstable homotopy theory.
He started publishing in 1952. Many of his early papers are concerned with the study of Whitehead product
s and their behaviour under suspension and more generally with the (unstable) homotopy groups of spheres
. In a 1957 paper he showed the first non-existence result for the Hopf invariant
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
(which he calls the toric construction) and the Toda fibration, 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
) 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
s.
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
ese mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, who specializes in stable
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
and unstable homotopy theory.
He started publishing in 1952. Many of his early papers are concerned with the study of Whitehead product
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 .- Definition :Given elements f \in \pi_k, g \in \pi_l, the Whitehead bracket...
s and their behaviour under suspension and more generally with the (unstable) homotopy groups of spheres
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 of spheres viewed as topological spaces, forgetting...
. In a 1957 paper he showed the first non-existence result for the Hopf invariant
Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.- Motivation :In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map\eta\colon S^3 \to S^2,...
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
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 more information....
(which he calls the toric construction) and the Toda fibration, 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
Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
(as modules over the Steenrod algebra
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 Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p...
) 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
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s.