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23-436: Whitehead's original homomorphism is defined geometrically, and gives a homomorphism of abelian groups for integers q , and r ≥ 2 {\displaystyle r\geq 2} . (Hopf defined this for the special case q = r + 1 {\displaystyle q=r+1} .) The J -homomorphism can be defined as follows. An element of the special orthogonal group SO( q ) can be regarded as

46-640: A Fields Medal at the International Congress of Mathematicians held in Helsinki . From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College , Oxford . Quillen retired at the end of 2006. He died from complications of Alzheimer's disease on April 30, 2011, aged 70, in Florida. Quillen's best known contribution (mentioned specifically in his Fields medal citation)

69-443: A ring , the endomorphism ring of G . For example, the endomorphism ring of the abelian group consisting of the direct sum of m copies of Z / n Z is isomorphic to the ring of m -by- m matrices with entries in Z / n Z . The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category ; the existence of direct sums and well-behaved kernels makes this category

92-702: A homomorphism can be interpreted as measuring how close it is to being an isomorphism. The first isomorphism theorem states that the image of a group homomorphism, h ( G ) is isomorphic to the quotient group G /ker h . The kernel of h is a normal subgroup of G . Assume u ∈ ker ⁡ ( h ) {\displaystyle u\in \operatorname {ker} (h)} and show g − 1 ∘ u ∘ g ∈ ker ⁡ ( h ) {\displaystyle g^{-1}\circ u\circ g\in \operatorname {ker} (h)} for arbitrary u , g {\displaystyle u,g} : The image of h

115-405: A limit as q tends to infinity gives the stable J -homomorphism in stable homotopy theory : where S O {\displaystyle \mathrm {SO} } is the infinite special orthogonal group, and the right-hand side is the r -th stable stem of the stable homotopy groups of spheres . The image of the J -homomorphism was described by Frank Adams  ( 1966 ), assuming

138-522: A map Applying the Hopf construction to this gives a map in π r + q ( S q ) {\displaystyle \pi _{r+q}(S^{q})} , which Whitehead defined as the image of the element of π r ( SO ⁡ ( q ) ) {\displaystyle \pi _{r}(\operatorname {SO} (q))} under the J-homomorphism. Taking

161-429: A map and the homotopy group π r ( SO ⁡ ( q ) ) {\displaystyle \pi _{r}(\operatorname {SO} (q))} ) consists of homotopy classes of maps from the r -sphere to SO( q ). Thus an element of π r ( SO ⁡ ( q ) ) {\displaystyle \pi _{r}(\operatorname {SO} (q))} can be represented by

184-534: A map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous. Let e H {\displaystyle e_{H}} be the identity element of the ( H , ·) group and u ∈ G {\displaystyle u\in G} , then Now by multiplying for the inverse of h ( u ) {\displaystyle h(u)} (or applying

207-576: A number of years at several other universities. He visited France twice: first as a Sloan Fellow in Paris, during the academic year 1968–69, where he was greatly influenced by Grothendieck , and then, during 1973–74, as a Guggenheim Fellow . In 1969–70, he was a visiting member of the Institute for Advanced Study in Princeton , where he came under the influence of Michael Atiyah . In 1978, Quillen received

230-421: Is k ∘ h  : G → K . This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category (specifically the category of groups ). If G and H are abelian (i.e., commutative) groups, then the set Hom( G , H ) of all group homomorphisms from G to H is itself an abelian group: the sum h + k of two homomorphisms is defined by The commutativity of H

253-425: Is 0 or 1 mod 8 and positive, the order of the image is 2 (so in this case the J -homomorphism is injective ). If r is 3 or 7 mod 8, the image is a cyclic group of order equal to the denominator of B 2 n / 4 n {\displaystyle B_{2n}/4n} , where B 2 n {\displaystyle B_{2n}} is a Bernoulli number . In the remaining cases where r

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276-599: Is 2, 4, 5, or 6 mod 8 the image is trivial because π r ( SO ) {\displaystyle \pi _{r}(\operatorname {SO} )} is trivial. Michael Atiyah  ( 1961 ) introduced the group J ( X ) of a space X , which for X a sphere is the image of the J -homomorphism in a suitable dimension. The cokernel of the J -homomorphism J : π n ( S O ) → π n S {\displaystyle J\colon \pi _{n}(\mathrm {SO} )\to \pi _{n}^{S}} appears in

299-502: Is a subgroup of H . The homomorphism, h , is a group monomorphism ; i.e., h is injective (one-to-one) if and only if ker( h ) = { e G }. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection: forms a group under matrix multiplication. For any complex number u the function f u  : G → C defined by If h  : G → H and k  : H → K are group homomorphisms, then so

322-450: Is needed to prove that h + k is again a group homomorphism. The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Hom( K , G ) , h , k are elements of Hom( G , H ) , and g is in Hom( H , L ) , then Since the composition is associative , this shows that the set End( G ) of all endomorphisms of an abelian group forms

345-510: The Adams conjecture of Adams (1963) which was proved by Daniel Quillen  ( 1971 ), as follows. The group π r ( SO ) {\displaystyle \pi _{r}(\operatorname {SO} )} is given by Bott periodicity . It is always cyclic ; and if r is positive, it is of order 2 if r is 0 or 1 modulo 8, infinite if r is 3 or 7 modulo 8, and order 1 otherwise ( Switzer 1975 , p. 488). In particular

368-402: The Adams conjecture , formulated by Frank Adams , in homotopy theory . His proof of the conjecture used techniques from the modular representation theory of groups , which he later applied to work on cohomology of groups and algebraic K -theory. He also worked on complex cobordism , showing that its formal group law is essentially the universal one. In related work, he also supplied

391-462: The cancellation rule) we obtain Similarly, Therefore for the uniqueness of the inverse: h ( u − 1 ) = h ( u ) − 1 {\displaystyle h(u^{-1})=h(u)^{-1}} . We define the kernel of h to be the set of elements in G which are mapped to the identity in H and the image of h to be The kernel and image of

414-413: The group Θ n of h -cobordism classes of oriented homotopy n -spheres ( Kosinski (1992) ). Group homomorphism In mathematics , given two groups , ( G ,∗) and ( H , ·), a group homomorphism from ( G ,∗) to ( H , ·) is a function h  : G → H such that for all u and v in G it holds that where the group operation on the left side of the equation is that of G and on

437-486: The image of the stable J -homomorphism is cyclic. The stable homotopy groups π r S {\displaystyle \pi _{r}^{S}} are the direct sum of the (cyclic) image of the J -homomorphism, and the kernel of the Adams e-invariant ( Adams 1966 ), a homomorphism from the stable homotopy groups to Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } . If r

460-545: The prototypical example of an abelian category . Daniel Quillen Daniel Gray Quillen (June 22, 1940 – April 30, 2011) was an American mathematician . He is known for being the "prime architect" of higher algebraic K -theory , for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978. From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford . Quillen

483-409: The right side that of H . From this property, one can deduce that h maps the identity element e G of G to the identity element e H of H , and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure". In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means

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506-542: Was born in Orange, New Jersey , and attended Newark Academy . He entered Harvard University , where he earned both his AB , in 1961, and his PhD in 1964; the latter completed under the supervision of Raoul Bott , with a thesis in partial differential equations . He was a Putnam Fellow in 1959. Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. He also spent

529-469: Was his formulation of higher algebraic K -theory in 1972. This new tool, formulated in terms of homotopy theory, proved to be successful in formulating and solving problems in algebra, particularly in ring theory and module theory. More generally, Quillen developed tools (especially his theory of model categories ) that allowed algebro-topological tools to be applied in other contexts. Before his work in defining higher algebraic K -theory, Quillen worked on

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