Ex1. Let f : S

USTC, School of Mathematical Sciences Winter semester 2018/12/06 Algebraic topology by Prof. Mao Sheng Exercise sheet 10

MA04311 Tutor: Lihao Huang, Han Wu 10 points

Posted online by Dr. Muxi Li 2 points for a problem

Ex1. Let f : S

→ S

be a map of degree zero. Show that there exist points x, y ∈ S

with f (x) = x and f (y) = y. Use this to show that if F is a continuous vector field defined on the unit ball D

in R

such that F (x) ̸= 0 for all x, then there exists a point on ∂D

where F points radially outward and another point on ∂D

where F points radially inward.

Ex2. A map f : S

→ S

satisfying f (x) = f ( −x) for all x is called an even map. Show that an even map S

→ S

must have even degree, and that the degree must in fact be zero when n is even. When n is odd, show there exist even maps of any given even degree. [Hints: If f is even, it factors as a composition S

→ RP

→ S

. Using the calculation of H

( RP

) in the text, show that the induced map H

(S

) → H

( RP

) sends a generator to twice a generator when n is odd. It may be helpful to show that the quotient map RP

→ RP

/ RP

induces an isomorphism on H

when n is odd.]

Ex3. Show the isomorphism between cellular and singular homology is natural in the following sense: A map f : X → Y that is cellular − satis- fying f (X

) ⊂ Y

for all n − induces a chain map f

between the cellular chain complexes of X and Y , and the map f

: H

(X) → H

(Y ) induced by this chain map corresponds to f

: H

(X) → H

(Y ) under the isomorphism H

≈ H

.

Ex4. What happens if one defines homology groups h

(X; G) as the ho- mology groups of the chain complex

· · · → Hom(G, C

(X)) → Hom(G, C

(X)) → · · · ?

More specifically, what are the groups h

(X; G) when G = Z, Z

, and Q?

Ex5. Let X be a Moore space M ( Z

, n) obtained from S

by attach- ing a cell e

by a map of degree m.

(a) Show that the quotient map X → X/S

= S

induces the trivial map on e H

( −; Z) for all i, but not on H

( −; Z). Deduce that the splitting in the universal coefficient theorem for cohomology cannot be natural.

(b) Show that the inclusion S

, → X induces the trivial map on e H

(; Z) for all i, but not on H

( −; Z).

Please hand in this homework on 12th Dec. 2018.

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