Stage 4 Topology

Topology:
See Geometry & Topology Monographs for further study. Also consult Allen's list of recommended books on topology [.pdf].

In general:

• Hocking J.G. and Young G.S. Topology
• Kelley J.L. General Topology - "What every young analyst should know" was the originally intended title. Morse-Kelley set theory was introduced in this book.
• Steenrod N. The Topology of Fibre Bundles
• Stillwell J. Classical Topology and Combinatorial Group Theory

Geometric Topology:

• Firby P.A. and Gardiner C.F. Surface Topology - One quote by Ian Stewart is sufficient. "This book is in the top ten for books conveying the dreadful secret that mathematics is fascinating."
• Gay D. Explorations in Topology: Map Coloring, Surfaces and Knots
• Ranicki A. History of Knot Theory (FREE!) - Contains dozens of historical material.
• Roberts J.D. Knots Knotes [.pdf] (FREE!)

Algebraic Topology:

• Bott R. and Tu L.W. Differential Forms in Algebraic Topology
• Dold A. Lectures on Algebraic Topology
• Eilenberg S. and Steenrod N.E. Foundations of Algebraic Topology
• Fulton W. Algebraic Topology: A First Course
• Hatcher A. Algebraic Topology (here for FREE!) - Very beautiful book.
• Lefschetz S. Algebraic Topology (FREE!) - Lefschetz developed new result in this monograph.
• Maunder C. R. F. Algebraic Topology
• Massey W.S. A Basic Course in Algebraic Topology
• Massey W.S. Algebraic Topology: An Introduction
• Mislin G. Algebraic Topology [.pdf] (FREE!)
• Moller J.M. Algebraic Topology Notes (FREE!)
• Munkres J.R. Elements of Algebraic Topology
• Spanier E.H. Algebraic Topology
• Switzer R.M. Algebraic Topology - Homology and Homotopy
• Vick J.W. Homology Theory: An Introduction to Algebraic Topology
• Wallace A.H. Algebraic Topology: Homology and Cohomology
• Wallace A.H. An Introduction to Algebraic Topology
• Whitehead G.H. Elements of Homotopy Theory
• Wilkins D.R. Algebraic Topology (FREE!)

K-theory:

• Hatcher A. Vector Bundles and K-theory (FREE!)
• Karoubi M. K-theory: An Introduction
• Rosenberg J. Algebraic K-Theory and Its Applications
• Srinivas V. Algebraic K-Theory
• Weibel C. The K-book: An Introduction to Algebraic K-theory (FREE!)

Differential Topology:

• Brin M.G. Introduction to Differential Topology [.pdf] or [.ps] here (FREE!)
• Guillemin V. and Pollack A. Differential Topology
• Hirsch M.W. Differential Topology
• Kosinski A.A. Differential Manifolds - An appendix by John W. Morgan on the work of Grigory Perelman is included in the 2007 Dover edition.
• Lang S. Introduction to Differentiable Manifolds
• Milnor J. Morse Theory - More advanced work, influential. Notes taken by M. Spivak and R. Wells.
• Milnor J.W. Topology from the Differentiable Viewpoint - As an introduction.
• Wallace A.H. Differential Topology: First Steps
• 張筑生：《微分拓樸》