As the title indicates, I'm trying to find books where the exposition of the main course of thought is done entirely or mostly in outlines of proofs, or as exercises with or without hints. I'm trying to force my reading to be more "active", and I think that such a book would be good training-wheels.

No particular topics, but preferably something on the introductory level. I'm particularly interested in basic Differential Geometry and/or Algebraic Topology related topics right now. (I have "undergraduate level" background in Real Analysis, Abstract Algebra, Linear Algebra and General Topology, and I'm trying to get started with Algebraic Topology and Differential Geometry for my own personal and educational enrichment.)

Texts on other topics would be welcome for later reference.

Much thanks.

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    $\begingroup$ I'm not sure if this is what you're looking for, but I'm reading Hatcher's Algebraic Topology right now, and I very often wish he had provided more detail in the proofs. So although it's frustrating for me, that might be right up your alley! $\endgroup$ – Eric Auld Aug 2 '13 at 8:54
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    $\begingroup$ This has been asked on MO: mathoverflow.net/questions/12709/… $\endgroup$ – user314 Aug 2 '13 at 18:26
  • $\begingroup$ Yaglom and Boltyanskii's Convex Figures is nice, and has fairly advanced material (from my perspective) such as selection theorems. Halmos has A Hilbert Space Problem Book. These were mentioned in the MO link, but I'm giving them thumbs up as well. $\endgroup$ – Trurl Aug 13 '13 at 12:33

One excellent algebraic topology book that leaves almost all the work to the reader is Modern Classical Homotopy Theory by Jeff Strom. To quote from the introduction:

Many authors of textbooks assert that the only way to learn the subject is to do the exercices. I have taken this to heart, and so there are no outright proofs in the book. Instead, theorems are followed by multi-part problems that guide the readers to find the proofs for themselves. To the expert, these problems will read as terse proofs, perhaps suitable for the exposition in a journal article. Reading this text, then, is a preparation for the experience of reading research articles. There are also a great many other problems incorporated into the main flow of the text, problems that develop interesting tangential results, explore applications, or carry out explicit calculations.

  • $\begingroup$ That's what I was going to suggest, too! $\endgroup$ – Kevin Carlson Aug 2 '13 at 9:28

The following are below your current level, but might interest you as examples of the genre, They are all by R.P. Burn. All proofs are in guided exercises.

The books are Groups: A Path to Geometry, and similarly titled books in number theory and (introductory) analysis.


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