Brouwer fixed point theorem (1) (p. 120)

Let I be a line segment, including its endpoints (I for Interval) and suppose that {f : I \rightarrow I} is a continuous endomap. Then this map must have a fixed point: a point x in I for which {f(x) = x}.

Brouwer fixed point theorem (2) (p. 121)

Let D be a closed disk (the plane figure consisting of all the points inside or on a circle), and f a continuous endomap of D. Then f has a fixed point.

Brouwer fixed point theorem (3) (p. 122)

Any continuous endomap of a solid ball has a fixed point.

Brouwer retraction theorem (I) (p. 122)

Consider the inclusion map {j : E \rightarrow I} of the two-point set E as boundary of the interval I. There is no continuous map which is a retraction for j.

Brouwer retraction theorem (II) (p. 123)

Consider the inclusion map {j : C \rightarrow D} of the circle C as boundary of the disk D into the disk. There is no continuous map which is a retraction for j.

Brouwer retraction theorem (III) (p. 123)

Consider the inclusion {j : S \rightarrow B} of the sphere S as boundary of the ball B into the ball. There is no continuous map which is a retraction for j.

Exercise 1 (p. 126)

Let {j : C \rightarrow D} be, as before, the inclusion of the circle into the disk. Suppose that we have two continuous maps {D \xrightarrow{f} D} [and] {D \xrightarrow{g} D}, and that g satisfies {g \circ j = j}. Use the retraction theorem to show that there must be a point x in the disk at which {f(x) = g(x)}. (Hint: The fixed point theorem is the special case {g = 1_D}, so try to generalize the argument we used in that special case.)

Exercise 2 (p. 126)

Suppose that A is a 'retract' of X, i.e. there are maps {A \xrightarrow{s} X \xrightarrow{r} A} with {r \circ s = 1_A}. Suppose also that X has the fixed point property for maps from T, i.e. for every endomap {X \xrightarrow{f} X}, there is a map {T \xrightarrow{x} X} for which {f x = x}. Show that A also has the fixed point property for maps from T. (Hint: The proof should work in any category, so it should only use the algebra of composition of maps.)

Exercise 3 (p. 126)

Use the result of the preceding exercise, and the fact that the antipodal map has no fixed point, to deduce each retraction theorem from the corresponding fixed point theorem.

Axiom 1 (p. 128)

If T is any object in 𝑪, and {T \xrightarrow{a} A} and {T \xrightarrow{s} S} are maps satisfying {h a = j s}, then {p a = s}.

Theorem 1 (p. 128)

If {B \xrightarrow{\alpha} A} satisfies {h \alpha j = j}, then {p \alpha} is a retraction for j.

Corollary 1 (p. 128)

If {h \alpha = 1_B}, then {p \alpha} is a retraction for j.

Axiom 2 (p. 129)

If T is any object in 𝑪, and {T \xrightarrow{f} B} [and] {T \xrightarrow{g} B} are any maps, then either there is a point {\mathbf{1} \xrightarrow{t} T} with {f t = g t}, or there is a map {T \xrightarrow{\alpha} A} with {h \alpha = g}.

Theorem 2 (p. 129)

Suppose we have maps B \xrightarrow{f} B\\\ B \xrightarrow{g} B and {g j = j}, then either there is a point {\mathbf{1} \xrightarrow{b} B} with {f b = g b}, or there is a retraction for {S \xrightarrow{j} B}.

Corollary 2 (p. 129)

If {B \xrightarrow{f} B}, then either there is a fixed point for f or there is a retraction for {S \xrightarrow{j} B}.

Exercise 4 (p. 132)

(a) Express the restrictions given above on my travel and yours by equations involving composition of maps, introducing other objects and maps as needed.

(b) Formulate the conclusion that at some time we meet, in terms of composition of maps. (You will need to introduce the object \mathbf{1}.)

(c) Guess a stronger version of Brouwer’s fixed point theorem in two dimensions, by replacing E, I, and R by the circle, disk, and plane. (You can do it in three dimensions too, if you want.)

(d) Try to test your guess in (c); e.g. try to invent maps for which your conjectured theorem is not true.