Students have asked me few times if I could recommend them a book with solved problems in algebraic topology. Unfortunately, the only one that springs to mind is Terry Lawson's Topology: A Geometric Approach, but I'm not quite satisfied with the exercises it contains.

On the other hand, there are lots of good exercises in a number of books (like Hatcher's) but they don't have solutions written up.

So could you recommend me a book with good exercises in algebraic topology that also contains their solutions?

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    $\begingroup$ IIRC, Prasolov's two books on topology have solutions and hints for most of the exercises. $\endgroup$ – Steve D Jan 3 '11 at 23:19

Dear Jankir,

1) The book by Sergey V. Matveev

Lectures on Algebraic Topology, Sergey V. Matveev (EMS Series of Lectures in Mathematics, 2006)

contains about 10 pages of hints and solutions to its exercises. That's not a bad ratio since the body of the book is only 82 pages long! Despite its terseness the book is amazingly nourishing: it contains Wirtinger's presentation of the fundamental group of a knot, higher homotopy groups, Poincaré duality, the Lefschetz fixed point theorem,... all starting from scratch. Of course not everything is proved but the numerous exercises prove that there is real meat to feed on in that short tome.

2) The book by T.W. Gamelin and Robert E. Greene

Introduction to Topology , Dover (2nd ed. 1999)

devotes its second half to algebraic topology (the first half being about general topology). There are 13 pages of solutions just for the exercises of the algebraic topology part. Here also the material is more advanced than one could expect in an introductory book: higher homotopy groups, Jordan theorem, degree of maps ,... The price ? $9.15

3) The book by H. Sato

Algebraic Topology: An Intuitive Approach, AMS (1999)

is also a book starting from scratch and reaching fairly advanced subjects in 110 pages, while giving serious technical proofs on the way. The only book I know with solved exercises on spectral sequences!

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If you don't like "intiutive proofs" stay away from Sato. You can find by googling quite a good amount of solutions to the hatchers book at the websites of profs from other universities. An example is http://noether.uoregon.edu/~sadofsky/636/. There are many websites like this. However exercises to chapter 0 are rare, which is the chapter that I find really unsufficient. To complement that, read Creating New Spaces from Old from Lee's Topological Manifolds. It makes an excellent complement to chapter 0.

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