Compute the homology of the one-point union of \(S^1 \times S^1\) and \(S^1\).
Solution
1. Reduced homology of a wedge of connected CW complexes is the direct sum of the reduced homologies of the summands.
This follows from the cellular chain complex of the wedge, or from Mayer–Vietoris using neighborhoods whose intersection is contractible.
2. The torus has \(H_0=H_2=\mathbb Z\) and \(H_1=\mathbb Z^2\), while \(S^1\) has \(H_0=H_1=\mathbb Z\).
These are the standard homology computations.
3. Therefore\[\boxed{H_k((S^1\times S^1)\vee S^1;\mathbb Z)\cong\begin{cases}\mathbb Z,&k=0,2,\\\mathbb Z^3,&k=1,\\0,&\text{otherwise.}\end{cases}}\]
Add the reduced homology groups in positive degrees and retain one copy of \(\mathbb Z\) in degree zero because the wedge is connected.