I'm concerned about the group structure on $[X,S^n]$, i.e. the set of homotopy classes of continuous maps from $X$ to $S^n$.


On the one hand, $[X,Y]$ has a group structure that is natural with respect to $X$ if and only if $Y$ is an H-space. The naturality is in the sense that $f:X'\to X$ induces a homomorphism $f\_{\star}:[X,Y]\to[X',Y]$. It's known that $S^n$ is an H-space only for $n=0,1,3,7$. Thus, for a general $X$, there's no natural group structure on $[X,S^n]$ when $n$ takes other values.


On the other hand, when $X$ is a closed smooth $d$-manifold, the Pontryagin-Thom construction establishes a bijection between $[X,S^n]$ and $\Omega\_{d-n}^{fr}(X)$, i.e. the (unstable) $(d-n)$th framed bordism set of $X$. When $d<2n-1$, two transverse $(d-n)$-submanifolds in $X$ have no intersection and the disjoint union induces a group structure on $\Omega\_{d-n}^{fr}(X)$. Then we can make $[X,S^n]$ into a group via the Pontryagin-Thom bijection.


Hence $[X,S^n]$ is a group when $X$ is a closed smooth manifold whose dimension $<2n-1$. I would like to know how "natural" or how "unnatural" this group structure is: (1) if $X'$ is a closed smooth manifold whose dimension $<2n-1$, does a smooth map $f:X'\to X$ induce a group homomorphism? (2) When $n=0,1,3,7$, does this group structure coincide with the H-space induced group structure?


**Edit 1** The question has been edited to correct a mistake pointed out by Gregory.


**Edit 2** As mentioned by Tyrone, $[X,S^n]$ is a cohomotopy group when $X$ is a complex of dimension $<2n-1$, with the multiplication induced by the folding map. It reduces to the H-space induced group if $n=0,1,3,7$. My **Q2** essentially asks whether *cohomotopy groups* coincide with *framed bordism groups* (when the latter applies).



The group structure on $[X,S^n]$ induced by the framed bordism
This is an answer to question (2). Let $n=0,1,3,7$ and $i=1,2$ and $d\leq 2n-2$. Let $e=(1,0,\ldots,0)\in S^n$


Let $f\_i:X\rightarrow S^n$ be two maps representing framed submanifolds $(M\_i,\nu\_i)$. Let $T\_i$ be tubular neighborhoods of $M\_i$.


By the assumptions on the dimensions, the maps can be chosen in such a way that


* $T\_1\cap T\_2=\emptyset$.
* The value $-e\in S^n$ is regular for both $f\_1,f\_2$
* $f\_i^{-1}(\{-e\})=M\_i$ and the framing induced by the differential is $\nu\_i$.
* $f\_i(X\setminus T\_i)=e$.


Let $g:X\rightarrow S^n$ be the product map $g(x)=f\_1(x)f\_2(x)$. By the choices made above $-e$ is a regular value and $g^{-1}(\{-e\})=M\_1\cup M\_2$. The induced framing on $M\_1\cup M\_2$ is $\nu\_1,\nu\_2$ on the respective components. This shows that the group structures coincide.


