# Fixed Points

## 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 basepoint-preserving 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 degree-one 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 degree-one 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?

Use the Brouwer fixed point theorem to show that an $$n \times n$$ matrix with nonnegative entries has a real eigenvalue.