Given $n$ circles $C_1, \dots, C_n$ glued by a common point $p$. Find the homology of the union $\cup_{i=1}^n C_i$.
I am defining $c_0 = \langle p \rangle = \mathbb{Z} \cdot p$ and $c_1 = \langle l_1, \dots, l_n \rangle = \mathbb{Z} \cdot l_1 + \dots + \mathbb{Z} \cdot l_n$.
$B_0 = \{0\}$ because $\partial p = 0$. $Z_0 = c_0$ because level $-1$ eats everything. Then $H_0 \cong \mathbb{Z}$.
$Z_1 = c_1$ because $\partial l_i = p - p = 0$ for $1 \leq i \leq n$.
But what about $B_1, H_1, Z_2$ and $H_2$? Intuitively I have circles one inside the other as petals of a rose and some in the opposite direction, but not so sure how to describe the elements of $c_2$, in particular, $\partial c_2$. Is it wrong to assume how they are distributed?
