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I do have one addition to make to the above. At our university we usually use a combination of Guillemin and Pollack and Milnor. There is another approach at a first course which some have found useful: Bott and Tu's book,

                  Differential forms in algebraic topology


This text covers an alternative set of topics that overlap both manifold theory and algebraic topology.

Disclaimers: (1) I have never used the text myself, but several colleagues have said in the past that it is a good book to use---and I am personally a big fan of Bott's approach to mathematical writing.

(2) If one uses Bott and Tu, then one has to sacrifice

                          transversality.


Andrew Ranicki once told me that transversality counts as one of the most important gems of last 20th century in mathematics.

3 deleted 37 characters in body

I do have one addition to make to the above. At our university we usually use a combination of Guillemin and Pollack and Milnor. There is another approach at a first course which some have found useful: Bott and Tu's book,

                  Differential forms in algebraic topology


This text covers an different alternative set of topics than Guillemin and Poolack and Milnor that overlaps overlap both manifold theory and algebraic topology.

Disclaimers: (1) I have never used the text myself, but several colleagues have said in the past that it is a good book to use---and I am personally a big fan of Bott's approach to mathematical writing.

(2) If one uses Bott and Tu, then one has to sacrifice

                          transversality.


Andrew Ranicki once told me that transversality counts as one of the most important gems of last century in mathematics

I do have one addition to make to the above. At our university we usually use a combination of Guillemin and Pollack and Milnor. There is another approach at a first course which some have found useful: Bott and Tu's book,

                  Differential forms in algebraic topology


This text covers an different set of topics than Guillemin and Poolack and Milnor that overlaps both manifold theory and algebraic topology.

Disclaimer

Disclaimers: (1) I have never used the text myself, but several colleagues have said in the past that it is a good book to use---and I am personally a big fan of Bott's approach to mathematical writing.

(2) If one uses Bott and Tu, then one has to sacrifice

                          transversality.


Andrew Ranicki once told me that transversality counts as one of the most important gems of last century in mathematics