Homology groups of a simplicial complex

Question

I have a question from a qualifying exam: let $X$ be the simplicial complex that consists of the 3-simplices $(v_1,v_2,v_3,v_4)$, $(v_3,v_4,v_5,v_6)$, $(v_1,v_2,v_5,v_6)$, where the $v_i$'s are all distinct 0-cells. The question asks to compute the homology groups of $X$.

Now, one can go ahead and compute the kernels and images of all the boundary maps, though this is sufficiently time consuming that I am inclined to believe that there is a trick or two that I am missing.

Thanks in advance for the help!

Answer 1

I think the circle is a deformation retract.