Ex 1. (1 pt) Define f : S

USTC, School of Mathematical Sciences Winter semester 2018/19 Algebraic topology by Prof. Mao Sheng Exercise sheet 2

MA04311 Tutor: Lihao Huang, Han Wu 9 points

Posted online by Dr. Muxi Li

Ex 1. (1 pt) Define f : S

× I → S

× I by f (θ, s) = (θ + 2πs, s), so f restricts to identity on the two bundary circles of S

× I. Show that f is homotopic to the identity by a homotopy f

that is stationary on one of the boundary circles, but not by any homotopy f

that is stationary on both boundary circles. [Consider what f does to the path s 7→ (θ

, s) for fixed θ

∈ S

.]

Ex 2. (1 pt) Does the Borsuk-Ulam theorem hold for the torus? In other words, for every map f : S

× S

→ R

must there exist (x, y) ∈ S

× S

such that f (x, y) = f (−x, −y)?

Ex 3. (1 pt) Let A

, A

, A

be compact sets in R

. Use the Borsuk-Ulam theorem to show that there is one plane P ⊂ R

that simultaneously divides each A

into two pieces of equal measure.

Ex 4. (1 pt) From the isomorphism π

(X × Y, (x

, y

)) ≈ π

(X, x

) × π

(Y, y

) it follows that loops in X × {y

} and {x

} × Y represent commuting elements of π

(X × Y, (x

, y

)). Construct an explicit homotopy demonstrating this.

Ex 5. (3 pts) Show that there are no retraction r : X → A in the following cases:

(a) X = R

with A any subspace homeomorphic to S

. (b) X = S

× D

with A its boundary torus S

× S

. (c) X = S

× D

and A the circle shown in the figure.

(d) X = D

∨ D

with A its boundary S

∨ S

.

(e) X a disk with two points on its boundary identified and A its boundary S

∨ S

. (f ) X the M¨ obius band and A its boundary circle.

Ex 6. (2 pts) Using the technique in the proof of Proposition 1.14, show that if a space X is obtained from a path-connected subspace A by attaching a cell e

with n ≥ 2, then the inclusion A ,→ X induces a surjection on π

. Apply this to show:

(a) The wedge sum S

∨ S

has fundamental group Z.

(b) For a path-connected CW complex X the inclusion map X

,→ X of its 1-skeleton induces a surjection π

(X

) → π

(X). [For the case that X has infinitely many cells, see Proposition A.1 in Appendix.]

Note: Please hand in this homework on next Wednesday (26

Sep. 2018).

1