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I'll answer your question in two steps. (1) You can make a degree one map $f : S^7 \to M$ a homeomorphism, and $C^\infty$-smooth on the complement of a point.

The idea is that you can construct $M$ as the union of two $D^7$'s

$$M = D^7 \sqcup_{h} D^7$$

where the gluing map $h : S^6 \to S^6$ is some diffeomorphisms of $S^6$ that does not extend to a diffeomorphism of $D^7$.

To get a map $S^7 \to M$, just write $S^7 = D^7 \sqcup_{Id_{S^6}} D^7$, the map on the first $D^7$ factor $S^7 \to M$ is just the identity, and on the 2nd $D^7$ factor you're coning-off $h$, i.e. the extension of $h : S^6 \to S^6$ to a homeomorphism of $D^7$ is given by $\tilde h(tv) = th(v)$ provided $v \in S^6$ and $t \in [0,1]$. This is usually called the Alexander trick, at least in the PL or topological categories. This map is smooth everywhere except $0$ in the 2nd $D^7$ factor.

(2) To fix this argument and get a smooth homeomorphism, replace $\tilde h(tv) = th(v)$ with $\tilde h(tv)=\beta(t)h(v)$ where $\beta : [0,1] \to [0,1]$ is a $C^\infty$-smooth homeomorphism with all derivatives of $\beta$ zero at $0$, but otherwise $\beta'(t) > 0$ for $t \in (0,1]$. You can cook up such functions readily using bump functions.

I think these arguments go back to Milnor's first papers on exotic smooth structures. You can make every representative of $\pi_7 M$ smooth, by iteratively suspending the degree $k$ map $S^1 \to S^1$ and appropriately "dampening" at the cone-points, such as the above construction.

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I'll answer your question in two steps. (1) You can make a degree one map $f : S^7 \to M$ a homeomorphism, and $C^\infty$-smooth on the complement of a point.

The idea is that you can construct $M$ as the union of two $D^7$'s

$$M = D^7 \sqcup_{h} D^7$$

where the gluing map $h : S^6 \to S^6$ is some diffeomorphisms of $S^6$ that does not extend to a diffeomorphism of $D^7$.

To get a map $S^7 \to M$, just write $S^7 = D^7 \sqcup_{Id_{S^6}} D^7$, the map on the first $D^7$ factor $S^7 \to M$ is just the identity, and on the 2nd $D^7$ factor you're coning-off $h$, i.e. the extension of $h : S^6 \to S^6$ to a homeomorphism of $D^7$ is given by $\tilde h(tv) = th(v)$ provided $v \in S^6$ and $t \in [0,1]$. This is usually called the Alexander trick, at least in the PL or topological categories. This map is smooth everywhere except $0$ in the 2nd $D^7$ factor.

(2) To fix this argument and get a smooth homeomorphism, replace $\tilde h(tv) = th(v)$ with $\tilde h(tv)=\beta(t)h(v)$ where $\beta : [0,1] \to [0,1]$ is a $C^\infty$-smooth homeomorphism with all derivatives of $\beta$ zero at $0$, but otherwise $\beta'(t) > 0$ for $t \in (0,1]$. You can cook up such functions readily using bump functions.

I think these arguments go back to Milnor's first papers on exotic smooth structures.