Computing the fundamental group of 1-skeletons

I’m aware of the idea of a CW complex and how to use the van Kampen, I’m struggling with the concept however of computing the fundamental group of the 1-skeleton of spaces.
For example, let us consider the simple spaces of the torus T\mathbb{T} and the Klein bottle K\mathbb{K}. The CW structures of these spaces consists of one 0-cell, two 1-cells and one 2-cell.

Hence we can determine the fundamental groups of these spaces simply by looking at the 1-skeleton, consisting of one 0-cell, two 1-cells and considering the attaching maps of the 2-cell φ:S1⟶X\varphi : S^1 \longrightarrow X.

I’m unsure of what this attaching map is and how to proceed with the computation of the fundamental group.



1 Answer


The generators are given by the 1-cells, the relations by the 2-cells. In the torus, you can attach a 2-cell to a square with opposite sides identified. The boundary of the 2-cell traces the path aba−1b−1aba^{-1}b^{-1}. So this is the relation imposed among the generators aa and bb.

The Klein bottle is similar.



Perfect! This is exactly what I was looking for. Thanks a lot
– Elliot
2 days ago