1 (Fall ’14) #topology/qual/work
Prove that, for every continuous map \(f : B^2 \to B^2\), there is a point \(x\) such that \(f (x) = x\).
This is the \(n = 2\) case of the Brouwer fixed point theorem; your proof shouldn’t appeal to either of the Brouwer or the Lefschetz fixed point theorems.
2 (Spring ’18) #topology/qual/work
Prove or disprove:
Every continuous map from \(S^2\) to \(S^2\) has a fixed point.
3 (Spring ’11) #topology/qual/work

State the Lefschetz Fixed Point Theorem for a finite simplicial complex \(X\).

Use degree theory to prove this theorem in case \(X = S^n\).
4 (Spring ’12) #topology/qual/work
a
Prove that for every continuous map \(f : S^2 \to S^2\) there is some \(x\) such that either \(f (x) = x\) or \(f (x) = x\).
Hint: Where \(A : S^2 \to S^2\) is the antipodal map, you are being asked to prove that either \(f\) or \(A \circ f\) has a fixed point.
b
Exhibit a continuous map \(f : S^3 \to S^3\) such that for every \(x \in S^3\), \(f (x)\) is equal to neither \(x\) nor \(x\).
Hint: It might help to first think about how you could do this for a map from \(S^1\) to \(S^1\).
5 (Spring ’14) #topology/qual/work
Show that a map \(S^n \to S^n\) has a fixed point unless its degree is equal to the degree of the antipodal map \(a : x \to x\).
6 (Spring ’08) #topology/qual/work
Give an example of a homotopy class of maps of \(S^1 \lor S^1\) each member of which must have a fixed point, and also an example of a map of \(S^1 \lor S^1\) which doesn’t have a fixed point.
7 (Spring ’17) #topology/qual/work
Prove or disprove:
Every map from \({\mathbf{RP}}^2 \lor {\mathbf{RP}}^2\) to itself has a fixed point.
8 (Fall ’09) #topology/qual/work
Find all homotopy classes of maps from \(S^1 \times {\mathbb{D}}^2\) to itself such that every element of the homotopy class has a fixed point.
9 (Spring ’10) #topology/qual/work
Let \(X\) and \(Y\) be finite connected simplicial complexes and let \(f : X \to Y\) and \(g : Y \to X\) be basepointpreserving maps.
Show that no matter how you homotope \(f \lor g : X \lor Y \to X \lor Y\), there will always be a fixed point.
10 (Fall ’12) #topology/qual/work
Let \(f = \operatorname{id}_{{\mathbf{RP}}^2} \lor \ast\) and \(g = \ast \lor id_{S^1}\) be two maps of \({\mathbf{RP}}^2 \lor S^1\) to itself where \(\ast\) denotes the constant map of a space to its basepoint.
Show that one map is homotopic to a map with no fixed points, while the other is not.
11 (Spring ’09) #topology/qual/work
View the torus \(T\) as the quotient space \({\mathbf{R}}^2 /{\mathbf{Z}}^2\).
Let \(A\) be a \(2 \times 2\) matrix with \({\mathbf{Z}}\) coefficients.
a
Show that the linear map \(A : {\mathbf{R}}^2 \to {\mathbf{R}}^2\) descends to a continuous map \({\mathcal{A}}: T \to T\).
b
Show that, with respect to a suitable basis for \(H_1 (T ; {\mathbf{Z}})\), the matrix \(A\) represents the map induced on \(H_1\) by \({\mathcal{A}}\).
c
Find a necessary and sufficient condition on \(A\) for \({\mathcal{A}}\) to be homotopic to the identity.
d
Find a necessary and sufficient condition on \(A\) for \({\mathcal{A}}\) to be homotopic to a map with no fixed points.
12 (Spring ’19) #topology/qual/work
a
Use the Lefschetz fixed point theorem to show that any degreeone map \(f : S^2 \to S^2\) has at least one fixed point.
b
Give an example of a map \(f : {\mathbf{R}}^2 \to {\mathbf{R}}^2\) having no fixed points.
c
Give an example of a degreeone map \(f : S^2 \to S^2\) having exactly one fixed point.
13 (Fall ’10) #topology/qual/work
For which compact connected surfaces \(\Sigma\) (with or without boundary) does there exist a continuous map \(f : \Sigma \to \Sigma\) that is homotopic to the identity and has no fixed point?
Explain your answer fully.
14 (Spring ’16) #topology/qual/work
Use the Brouwer fixed point theorem to show that an \(n \times n\) matrix with nonnegative entries has a real eigenvalue.