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In D. Freed's notes eqn (5.32), he defines the $J$-homomorphism geometrically by considering the equatorial $n$-sphere as an $n$-submanifold of $S^m$, and giving it a framing that makes it null-bordant, then he claims that restricting to pointed maps $g: S^n \to O(q)$ we obtain a homomorphism $J: [S^n,O(q)]_* \to \Omega_{n;S^m}^{fr}$. He does not explicitly define the homomorphism but just gives the domain and codomain. He does not explicitly say the operation of the homotopy group $[S^n,O(q)]_*$ that makes $J$ a homomorphism either.

Could somebody help me to fill in the missing details?

PhysicsMath
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1 Answers1

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My suggestion is to consider the case $n = 1$, $q = 2$, which you can visualize if you replace $S^3$ with affine space $\mathbb{A}^3$. The group operation in bordism is disjoint union, but that is equivalent to the connected sum, which you can approximate by the singular $1$-point union at the basepoint. In this case, you get a figure $8$. The group law of homotopy classes of based maps into $\text{O}(n)$ can be computed either by composition in $\text{O}(n)$ or by the usual product of higher homotopy groups, which uses the "co-$H$-space" structure of the sphere. That is the $1$-point union which I wrote about above. So I think if you consider that $1$-point union, you should be in good shape.

Brian Ng
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  • many thanks for the hint! I am correct you suggest visualizing the case of $n = 1, q = 2, m = 3$. And the $J$-homomorphism is defined by sending a based map $f$ representing an element of $\pi_1(O(2))$ (by the way by $O(n)$ what you really mean is $O(q)$ am I right?) to its image in $S^m$ which is a bordism of basepoint to itself. Moreover the group operation of the homotopy classes are equivalent to the singular 1-point union at the basepoint. – PhysicsMath Jul 29 '16 at 00:34
  • But I have two further questions: (1) how to show this homomorphism is well-defined i.e. if $f \sim g$ then $f(S^1) \sim g(S^1)$ as equivalent bordisms? (2) how to understand geometrically the use of $O(q)$? why can't we replace it with other group say $O^+(q)$ or even $SO(q)$? – PhysicsMath Jul 29 '16 at 00:34