Smooth representatives for elements of $\pi_7(\text{exotic$S^7$})$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:28:54Z http://mathoverflow.net/feeds/question/93483 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93483/smooth-representatives-for-elements-of-pi-7-textexotic-s7 Smooth representatives for elements of $\pi_7(\text{exotic$S^7$})$ Mariano Suárez-Alvarez 2012-04-08T07:22:17Z 2012-04-08T23:05:32Z <p>Let $M$ be $S^7$ with an exotic smooth structure. Since one can smoothen maps, there exist smooth maps $f:S^7\to M$ which are homotopic to the identity (relative to a base point, if you want). </p> <blockquote> <p>Can one make explicit one such map? Can such a map be an homeomorphism?</p> </blockquote> <p><strong>Little addendum.</strong> The smooth homeomorphism constructed in Ryan's answer below is of course not a diffeomorphism. </p> <blockquote> <p>Does one have some control on the non-smooth locus of the inverse of smooth homeomorphims, or on the type of their non-smoothness there?</p> </blockquote> <p>The inverse of Ryan's map is non-smooth only at the bad pole and I guess the initial map $h$ arises as the "conical differential" of the map there, so the singularity there is pretty bad. Maybe one can find other smooth homeomorphisms whose inverse has a larger non-smooth locus but with tamer non-smoothness there?</p> http://mathoverflow.net/questions/93483/smooth-representatives-for-elements-of-pi-7-textexotic-s7/93488#93488 Answer by Ryan Budney for Smooth representatives for elements of $\pi_7(\text{exotic$S^7$})$ Ryan Budney 2012-04-08T07:57:54Z 2012-04-08T08:24:37Z <p>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. </p> <p>The idea is that you can construct $M$ as the union of two $D^7$'s</p> <p>$$M = D^7 \sqcup_{h} D^7$$</p> <p>where the gluing map $h : S^6 \to S^6$ is some diffeomorphisms of $S^6$ that does not extend to a diffeomorphism of $D^7$. </p> <p>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. </p> <p>(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. </p> <p>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. </p> http://mathoverflow.net/questions/93483/smooth-representatives-for-elements-of-pi-7-textexotic-s7/93506#93506 Answer by Vitali Kapovitch for Smooth representatives for elements of $\pi_7(\text{exotic$S^7$})$ Vitali Kapovitch 2012-04-08T15:57:14Z 2012-04-08T16:30:23Z <p>Duràn wrote down an explicit formula for such map in <a href="http://www.springerlink.com/content/v775g60776g6m813/" rel="nofollow">"Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of $S^6$"</a>. That is he wrote an explicit formula for an exotic diffeomorphism from $S^6$ to $S^6$ which is homotopic but not isotopic to the identity. This the produces an explicit homeomorphism from $S^7$ to an exotic sphere by glueing as described by Ryan in his answer.</p> <p>Geometric properties of that particular map were later studied by various people. For example, it's written down explicitly on page 1 in <a href="http://www.ime.usp.br/~eufrasio/matcont/pt/edicoes.php?idvolume=29" rel="nofollow">"Bootstrapping $Ad$-Equivariant Maps, Diffeomorphisms and Involutions"</a> by Duràn and Rigas and Sperança (this link is freely accessible unlike the first one).</p>