Is there any structure in general on the $p$th cohomotopy set? The $p$th cohomotopy set of a pointed space $X$, $\pi^p(X)$, is the set of maps from $X$ to $S^p$ mod pointed homotopy. Unlike homotopy, there need be no natural group structure on $\pi^p(X)$ unless ($X$ is particularly nice or) $p\in\{0,1,3,7\}$: if $p$ is one of these values then we can use the group structure on $S^p$ to "add maps pointwise" (EDIT: it's not a group at $p=7$, since octonion multiplication isn't associative, but it's still algebraically nontrivial) but otherwise no such group structure exists.
At this point it's natural to ask whether we can still find some algebraic structure on $\pi^p(X)$ if $p\not\in\{0,1,3,7\}$ (and $X$ is not particularly nice). Unfortunately, Adams' theorem can be strengthened: Walter Taylor showed that in a precise sense there is no nontrivial ("demanding") algebraic structure on $S^p$ at all for $p\not\in\{0,1,3,7\}$. 
My question is whether this is lethal:

Is there some $p\not\in \{0,1,3,7\}$ such that there exists some natural algebraic structure on $\pi^p(X)$ for arbitrary $X$ (or at least for $X$ substantially more general than suspensions)?

By "algebraic structure" I really intend to cast a wide net, hence the universal algebra tag. One possible precisiation of the question would be the following: letting $\tau$ be the "obvious" topology on $\pi^p(X)$, is there in the sense of Taylor's paper a nontrivial algebraic structure on $\pi^p(X)$ compatible with $\tau$? However, I'm also open to other reasonable interpretations.
 A: I am surprised that the wikipedia article does not mention the cohomotopy groups which were invented by Borsuk in 1936. They are not defined for arbitrary $X$, but if $X$ is a CW-complex of dimension $\le 2n-2$, then you get the structure of an abelian group on $\pi^n(X)$. This is also true for any metric space of dimension $\le 2n-2$.
See
Borsuk, K. "Sur les groupes des classes de transformations continues." CR Acad. Sci. Paris 202.1400-1403 (1936): 2.
Borsuk, Karol. Theory of retracts. Vol. 44. Państwowe Wydawn. Naukowe, 1967. [Here Chapter II Section 11.]
Hilton, Peter. "On some contributions of Karol Borsuk to homotopy theory." Topological Methods in Nonlinear Analysis 1.1 (1993): 9-14.
https://www.tmna.ncu.pl/static/files/v01n1-03.pdf
Spanier, E.H. "Borsuk's cohomotopy groups." Annals of Mathematics (1949): 203-245.
https://www.jstor.org/stable/1969362?seq=1
Borsuk, Karol. "On a generalization of the cohomotopy groups." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys 8 (1960): 615-620.
