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This collection of explicit calculations in algebraic topology is written for students interested in symbolic calculations. In particular, if you need help in finding some material for explicit symbolic calculations to exercise your computational skills, this short book is for you. Here you will find various calculations in the quotient rings of multivariate polynomial rings due to some constructions in algebraic topology. Therefore, if you want to contribute to computer algebra and write some code or a library, most likely you will find some of the tasks suggested in the final section.