Homology of the simplicial complex obtained from an octahedron by removing 4 faces.

Question

Suppose you have the surface of an octahedron and you remove 4 of the eight faces as follows:

If you remove one face then you don't remove all the adyacent faces and so on.

You can look at this as a simpicial complex so you should be able to calculate it's simpicial homology, but that's where I'm stuck.

I suspect it has the same or similar homology as a two torus but I´m not sure.

http://paulscottinfo.ipage.com/polyhedra/platonic/octahedron.html

For example, in the image I linked, you would remove the red and blue faces (or the yellow and green) and the two faces which would be opposite to the blue and red faces.

Answer 1

You could write down what the faces of the simplicial complex are and the compute boundary maps and homology. This is a useful exercise if you are new to simplicial topology.

Or you could just look at that picture and imagine poking out the red and blue faces. Both the top and bottom half are homotopic to a circle. You get that $\tilde{H}_1$ has dimension 2, and all other reduced homologies vanish.