Hatcher-Algebraic-Topology · Problem 18

The Fundamental Group

Mathematics — Graph Theory · attaching-cells-surjection-pi1

Problem

Using Lemma 1.15, show that if a space

X

is obtained from a path-connected subspace

A

by attaching a cell

e^n

with

n \geq 2

, then the inclusion

A \hookrightarrow X

induces a surjection on

\pi_1

. Apply this to show: ((a)) The wedge sum

S^1 \vee S^2

has fundamental group

\mathbb{Z}

. ((b)) For a path-connected CW complex

X

the inclusion map

X^1 \hookrightarrow X

of its 1-skeleton induces a surjection

\pi_1(X^1) \to \pi_1(X)

. [For the case that

X

has infinitely many cells, see Proposition A.1 in the Appendix.]