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J. P. May's superb book, "A Concise Course in Algebraic Topology," starts with a great deal on homotopy theory, and doesn't really get to homology until nearly half way through. I learned a great deal from this approach, and think that it is the best way to teach algebraic topology. But May's book is rather probably too difficult for a "first course" in algebraic topology.

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J. P. May's superb book, "A Concise Course in Algebraic Topology," starts with a great deal on homotopy theory, and doesn't really get to homology until nearly half way through. I learned a great deal from this approach, and think that it is the best way to teach algebraic topology. But May's book is rather too difficult for a "first course" in algebraic topology.