I am trying to compute homology and fundamental group of the following simplicial complex, $K$, and perhaps above all gain some intuition:
The shades areas correspond to facets. I.e. it is a simplicial complex on vertices $\{a,b,c,d,e,f\}$ with facets $[a,b,c],[c,d,e],[a,d,f],[b,e,f]$ (the edges are obviously decided by this).
What I want to do is:
Compute the homology groups of $K$ by being blunt and computing images and kernels of the boundary homomorphism.
Compute the fundamental group of $K$ using van Kampen's Theorem.
Ill include my progress:
1. \begin{align} H_2&= \ker \partial_2 / im \partial_3\\ H_1&= \ker \partial_1 / im\partial_2\\ H_0&= \ker \partial_0 / im\partial_1 \end{align}
Now $\ker \partial_0 \cong \mathbb{Z}^6$ since there are $6$ vertices (we now know that $im\partial_1 \cong \mathbb{Z}^5$ by connectedness, but lets ignore this and compute everything explicitly). Now is where I get lost. I have $8$ $1-$cycles and I don't even know how many $2-$cylces I have. How do I proceed computing the non-trivial images and kernels of $\partial$?
EDIT: OK, I got a bit further on 1.:
\begin{align} Im \partial_1 &= F\{b-a,c-b,d-c,e-d,e-c,f-e,e-b,f-b,f-d,f-a,d-a\}\cong\mathbb{Z}^5\\ \ker \partial 1 &=F\{\text{elementary 1-cycles}\}\\ &=F\{\text{#faces of a pyramid with bottom split in four}\}\cong \mathbb{Z}^8\\ Im\partial_2&= \end{align} 2.
We know that there are $4$ hollow triangels which each have fundamental $\mathbb{Z}$ if I retract the solid triangels to single points. So the fundamental group should be $\mathbb{Z}^4 / x$ where $x$ are some relations induced by the shared vertices. How do I proceed?
My "check" for 2.: I have constructed a spanning tree via the path: $a\to b \to c \to d \to e\to f$. Then, the remaining edges must be the generators, so we are left with $$ g_{ac},g_{cf},g_{df},g_{ad},g_{ae},g_{be},g_{bf}.$$
and it is fairly easy to see using the relations obtained from the facets that \begin{align} g_{ac}&=g_{ab}g_{bc}=1\\ g_{cf}&=g_{df}:=s\\ g_{ad}&=g_{ae}:=t\\ g_{be}&=g_{bf}:=u\\ \end{align} so that $s,t,u$ is what remains. Hence the fundamental group must be a free group on $3$ generators.