What is a basic, graduate-level introduction to algebraic topology? I think Hatcher is a great book, but I want to learn the subject from the point of view of simplicial complexes. Primarily, I want to be able to do basic computations (finding fundamental groups, Mayer–Vietoris sequences, van Kampen theorem). More abstract things, like category theory, are not so important to me.
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Seifert and Threlfall's "A textbook of topology" is all with simplicial complexes. It's a very nice book. It has an excellent treatment of Poincare duality.
The purely simplicial perspective without much category theory is somewhat out of fashion but Seifert and Threlfall is one of the best references for that.
There is a recent book by Ferrario and Piccinini: "Simplicial Structures in Topology Simplicial Structures in Topology" that covers the basic material from a very geometric prespective. See http://www.springer.com/mathematics/geometry/book/978-1-4419-7235-4