Recommendation for Algebraic Topology References Supplementing Fulton

Question

I am about to start an algebraic topology course following Fulton's Algebraic Topology: A First Course. Going through the book, I find its approach somewhat idiosyncratic; unlike most such books it does not begin by building up singular homology or cell complexes but works through the de Rham theory in $\mathbb{R}^2.$ I am looking for other references that complement this book. Are there other algebraic topology books that take a similarly strong analytic approach? So far I've got Bott and Tu's Differential Forms in Algebraic Topology and John Roe's Winding Around. I can't even seem to find lecture notes based on this book.