This is an answer to question (2). Let $n=0,1,3,7$ and $i=1,2$ and $d\leq 2n-2$

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 $-1\in S^n$ is regular for both $f_1,f_2$
• $f_i^{-1}(\{-1\})=M_i$ and is framed by $\nu_i$.
• $f_i(X\setminus T_i)=1$.

Let $g:X\rightarrow S^n$ be the product map $g(x)=f_1(x)f_2(x)$. By the choices made above $-1$ is a regular value and $g^{-1}(\{-1\})=M_1\cup M_2$. 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.

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.

This is an answer to question (2). Let $n=0,1,3,7$ and $i=1,2$ and $d\leq 2n-2$

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 $-1\in S^n$ is regular for both maps and $f_i^{-1}(-1)=M_i$ and $f_i(X\setminus M_i)=1$.

Let $g:X\rightarrow S^n$ be the product map $g(x)=f_1(x)f_2(x)$. By the choices made above $-1$ is a regular value and $g^{-1}(-1)=M_1\cup M_2$. The induced framing on $M_1\cup M_2$ is $\nu_i$ on both components. This shows that the group structures coincide.

This is an answer to question (2). Let $n=0,1,3,7$ and $i=1,2$ and $d\leq 2n-2$

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 $-1\in S^n$ is regular for both $f_1,f_2$
• $f_i^{-1}(\{-1\})=M_i$ and is framed by $\nu_i$.
• $f_i(X\setminus T_i)=1$.

Let $g:X\rightarrow S^n$ be the product map $g(x)=f_1(x)f_2(x)$. By the choices made above $-1$ is a regular value and $g^{-1}(\{-1\})=M_1\cup M_2$. 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.