Topological Fixed Point Exercises

This is one of three sets of fixed point ex­er­cises. The first post in this se­quence is here, giv­ing con­text.

  1. (1-D Sperner’s lemma) Con­sider a path built out of edges as shown. Color each ver­tex blue or green such that the left­most ver­tex is blue and the right­most ver­tex is green. Show that an odd num­ber of the edges will be bichro­matic.

  1. (In­ter­me­di­ate value the­o­rem) The Bolzano-Weier­strass the­o­rem states that any bounded se­quence in has a con­ver­gent sub­se­quence. The in­ter­me­di­ate value the­o­rem states that if you have a con­tin­u­ous func­tion such that and , then there ex­ists an such that . Prove the in­ter­me­di­ate value the­o­rem. It may be helpful later on if your proof uses 1-D Sperner’s lemma and the Bolzano-Weier­strass the­o­rem.

  2. (1-D Brouwer fixed point the­o­rem) Show that any con­tin­u­ous func­tion has a fixed point (i.e. a point with ). Why is this not true for the open in­ter­val ?

  3. (2-D Sperner’s lemma) Con­sider a tri­an­gle built out of smaller tri­an­gles as shown. Color each ver­tex red, blue, or green, such that none of the ver­tices on the large bot­tom edge are red, none of the ver­tices on the large left edge are green, and none of the ver­tices on the large right edge are blue. Show that an odd num­ber of the small tri­an­gles will be trichro­matic.

  1. Color the all the points in the disk as shown. Let be a con­tin­u­ous func­tion from a closed tri­an­gle to the disk, such that the bot­tom edge is sent to non-red points, the left edge is sent to non-green points, and the right edge is sent to non-blue points. Show that sends some point in the tri­an­gle to the cen­ter.

  1. Show that any con­tin­u­ous func­tion from closed tri­an­gle to it­self has a fixed point.

  2. (2-D Brouwer fixed point the­o­rem) Show that any con­tin­u­ous func­tion from a com­pact con­vex sub­set of to it­self has a fixed point. (You may use the fact that given any closed con­vex sub­set of , the func­tion from to which pro­jects each point to the near­est point in is well defined and con­tin­u­ous.)

  3. Reflect on how non-con­struc­tive all of the above fixed-point find­ings are. Find a pa­ram­e­ter­ized class of func­tions where for each , , and the func­tion is con­tin­u­ous, but there is no con­tin­u­ous way to pick out a sin­gle fixed point from each func­tion (i.e. no con­tin­u­ous func­tion such that is a fixed point of for all ).

  4. (Sperner’s lemma) Gen­er­al­ize ex­er­cises 1 and 4 to an ar­bi­trary di­men­sion sim­plex.

  5. (Brouwer fixed point the­o­rem) Show that any con­tin­u­ous func­tion from a com­pact con­vex sub­set of to it­self has a fixed point.

  6. Given two nonempty com­pact sub­sets , the Haus­dorff dis­tance be­tween them is the supre­mum over all points in ei­ther sub­set of the dis­tance from that point to the other sub­set. We call a set val­ued func­tion a con­tin­u­ous Haus­dorff limit if there is a se­quence of con­tin­u­ous func­tions from to whose graphs, , con­verge to the graph of , , in Haus­dorff dis­tance. Show that ev­ery con­tin­u­ous Haus­dorff limit from a com­pact con­vex sub­set of to it­self has a fixed point (a point such that ).

  7. Let and be nonempty com­pact con­vex sub­sets of . We say that a set val­ued func­tion, is a Kaku­tani func­tion if the graph of , , is closed, and is con­vex and nonempty for all . For ex­am­ple, we could take and to be the in­ter­val , and we could have send each to , map to the whole in­ter­val , and map to . Show that ev­ery Kaku­tani func­tion is a con­tin­u­ous Haus­dorff limit. (Hint: Start with the case where is a unit cube. Con­struct by break­ing into small cubes of side length . Con­stuct a smaller cube of side length within each cube. Send each small to the con­vex hull of the images of all points in the cube with a con­tin­u­ous func­tion, and glue these to­gether with straight lines. Make sure you don’t ac­ci­den­tally get ex­tra limit points.)

  8. (Kaku­tani fixed point the­o­rem) Show that ev­ery Kaku­tani func­tion from a com­pact con­vex sub­set of to it­self has a fixed point.


Please use the spoilers fea­ture—the sym­bol ‘>’ fol­lowed by ‘!’ fol­lowed by space -in your com­ments to hide all solu­tions, par­tial solu­tions, and other dis­cus­sions of the math. The com­ments will be mod­er­ated strictly to hide spoilers!

I recom­mend putting all the ob­ject level points in spoilers and leav­ing meta­data out­side of the spoilers, like so: “I think I’ve solved prob­lem #5, here’s my solu­tion <spoilers>” or “I’d like help with prob­lem #3, here’s what I un­der­stand <spoilers>” so that peo­ple can choose what to read.