Recently I'm reading Donaldson's Geometry of four manifolds. It seems to me that the book requires a lot for background. Additionally, the proof in the book is too sketchy without too much detail. I had a really hard time to digest the content in the book. Do we have other textbook demonstrating the same topic with more detail and background?



About Simon Donaldson's book on four dimensional manifold
Please do not ignore the other author, Peter Kronheimer. Based on all of the material I've read, I do not agree with your belief about the book. I think it is more detailed than you will find elsewhere which covers *all* of that material. Here are some useful alternatives, though:


Take that book and replace the structure group $SU(2)$ by $SO(3)$. This route is taken in Petrie-Randall's **"Connections, definite forms, and four-manifolds"**.


If you care more for easy digestion and less about whether it is sketchy, the closest thing I can think of to Donaldson-Kronheimer's book is Freed-Uhlenbeck's **"Instantons and four-manifolds"**.


For rigorous yet digestive background, I recommend Booss-Bleecker's **"Topology and analysis: the Atiyah-Singer index formula and gauge-theoretic physics"**. Other relevent material, which has a lot of explanations/comments but not complete and not always rigorous, are Gompf-Stipsicz' **"4-manifolds and Kirby calculus"** and Scorpan's **"The wild world of 4-manifolds"**.


