Hatcher-Algebraic-Topology · Problem 3E.2

Cohomology

Mathematics — Graph Theory · bockstein-homomorphisms

Problem

In this problem we will derive one half of the classification of lens spaces up to homotopy equivalence, by showing that if

L_m(\ell_1, \cdots, \ell_n) \simeq L_m(\ell_1', \cdots, \ell_n')

then

\ell_1 \cdots \ell_n \equiv \pm \ell_1' \cdots \ell_n' k^n \mod m

for some integer

k

. The converse is Exercise 29 for \S4.2. (a) Let

L = L_m(\ell_1, \cdots, \ell_n)

and let

\mathbb{Z}_m^*

be the multiplicative group of invertible elements of

\mathbb{Z}_m

. Define

t \in \mathbb{Z}_m^*

by the equation

xy^{n-1} = tz

where

x

is a generator of

H^1(L; \mathbb{Z}_m)

y = \beta(x)

, and

z \in H^{2n-1}(L; \mathbb{Z}_m)

is the image of a generator of

H^{2n-1}(L; \mathbb{Z})

. Show that the image

\tau(L)

t

in the quotient group

\mathbb{Z}_m^*/\pm(\mathbb{Z}_m^*)^n

depends only on the homotopy type of

L

. (b) Given nonzero integers

k_1, \cdots, k_n

, define a map

\tilde{f}: S^{2n-1} \to S^{2n-1}

sending the unit vector

(r_1 e^{i\theta_1}, \cdots, r_n e^{i\theta_n})

\mathbb{C}^n

(r_1 e^{ik_1\theta_1}, \cdots, r_n e^{ik_n\theta_n})

. Show: ((i))

\tilde{f}

has degree

k_1 \cdots k_n

. ((ii))

\tilde{f}

induces a quotient map

f: L \to L'

for

L' = L_m(\ell_1', \cdots, \ell_n')

provided that

k_j \ell_j \equiv \ell_j' \mod m

for each

j

. ((iii))

f

induces an isomorphism on

\pi_1

, hence on

H^1(-; \mathbb{Z}_m)

. ((iv))

f

has degree

k_1 \cdots k_n

, i.e.,

f_*

is multiplication by

k_1 \cdots k_n

H_{2n-1}(-; \mathbb{Z})

. (c) Using the

f

in (b), show that

\tau(L) = k_1 \cdots k_n \tau(L')

. (d) Deduce that if

L_m(\ell_1, \cdots, \ell_n) \simeq L_m(\ell_1', \cdots, \ell_n')

, then

\ell_1 \cdots \ell_n \equiv \pm \ell_1' \cdots \ell_n' k^n \mod m

for some integer

k