Hatcher-Algebraic-Topology · Problem 29

Homotopy Theory

Mathematics — Graph Theory · lens-space-homotopy-classification

Problem

Finish the homotopy classification of lens spaces begun in Exercise 2 of §3.E by showing that two lens spaces

L_m(\ell_1, \cdots, \ell_n)

and

L_m(\ell'_1, \cdots, \ell'_n)

are homotopy equivalent if

\ell_1 \cdots \ell_n \equiv \pm k^n \ell'_1 \cdots \ell'_n \bmod m

for some integer

k

, via the following steps: (a) Reduce to the case

k=1

by showing that

L_m(\ell'_1, \cdots, \ell'_n) = L_m(k\ell'_1, \cdots, k\ell'_n)

k

is relatively prime to

m

. [Rechoose the generator of the

\mathbb{Z}_m

action on

S^{2n-1}

.] (b) Let

f: L \to L'

be a map constructed as in part (b) of the exercise in §3.E. Construct a map

g: L \to L'

as a composition

L \to L \vee S^{2n-1} \to L \vee S^{2n-1} \to L'

where the first map collapses the boundary of a small ball to a point, the second map is the wedge of the identity on

L

and a map of some degree

d

S^{2n-1}

, and the third map is

f

L

and the projection

S^{2n-1} \to L'

S^{2n-1}

. Show that

g

has degree

k_1 \cdots k_n + dm

, that is,

g

induces multiplication by

k_1 \cdots k_n + dm

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

. [Show first that a lift of

g

to the universal cover

S^{2n-1}

has this degree.] (c) If

\ell_1 \cdots \ell_n \equiv \pm \ell'_1 \cdots \ell'_n \bmod m

, choose

d

so that

k_1 \cdots k_n + dm = \pm 1

and show this implies that

g

induces an isomorphism on all homotopy groups, hence is a homotopy equivalence. [For

\pi_i

with

i > 1

, consider a lift of

g

to the universal cover.]