J-homomorphism

In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres.

Version

v1 · 2026-09-28 · History

Domain-specific #

10168

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Stable Homotopy Theory, Algebraic Topology → Mathematics

Core Idea

J-homomorphism is treated here as the recurring stable homotopy identity summarized by this source-grounded definition: In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of . and the homotopy group \pir(\operatorname{SO}(q)) ) consists of homotopy classes of maps from the r-sphere to SO(q). in \pi{r+q}(S^q) , which Whitehead.

Scope of Application

• Definition. J \colon \pir (\mathrm{SO}(q)) \to \pi{r+q}(S^q).

• Definition. An element of the special orthogonal group SO(q) can be regarded as a map.

• Definition. and the homotopy group \pir(\operatorname{SO}(q)) ) consists of homotopy classes of maps from the r-sphere to SO(q).

• Definition. Thus an element of \pir(\operatorname{SO}(q)) can be represented by a map.

• Applying the Hopf construction to this gives a map. S^{r+q}= S^rS) =S^q.}\rightarrow S( S^{q-1

Clarity

A clear use of J-homomorphism names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. The strongest recognition evidence in the frozen account is: The group \pir(\operatorname{SO}) is given by Bott periodicity.

Manages Complexity

J-homomorphism compresses multiple stable homotopy details into a stable diagnostic relation. The source shows both the central mechanism—thus an element of \pir(\operatorname{SO}(q)) can be represented by a map.—and the practical consequence—j \colon \pir (\mathrm{SO}(q)) \to \pi{r+q}(S^q). This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.

Abstract Reasoning

1. Type the carrier. Identify the stable homotopy entities to which the claim applies.
2. State the relation. Use the source-grounded identity: In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres.
3. Check operation and conditions. The image of the J-homomorphism was described by , assuming the Adams conjecture of which was proved by , as follows.
4. Demand recognition evidence. The group \pir(\operatorname{SO}) is given by Bott periodicity.
5. Test variation.

Knowledge Transfer

Within the home domain. Knowledge about J-homomorphism transfers literally when a new case preserves the same carrier type, relation, and recognition test. J \colon \pir (\mathrm{SO}(q)) \to \pi{r+q}(S^q). An element of the special orthogonal group SO(q) can be regarded as a map. Beyond the home domain. No canonical parent is asserted for J-homomorphism. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Neighborhood in Abstraction Space

J-homomorphism sits in a crowded region of the domain-specific corpus (37^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Structures, Groups & Operators (42 abstractions)

Nearest neighbors

• Steenrod problem — 0.89
• Hochschild homology — 0.89
• Supermodule — 0.88
• Character variety — 0.88
• Lefschetz zeta function — 0.87

Computed from structural-signature embeddings · 2026-10-08