Math 3760: Topics in Topology
Here is the first-day
- Differential Topology, V. Gullemin and A. Pollack. "Friendly"
and a nice read.
- Foundations of Differentiable Manifolds and Lie Groups, F.
Warner. Less friendly. Terse to a fault and lacking
examples in places, it also does everything "right" (a double-edged sword,
but very useful for referencing).
Algebraic Topology, Allen Hatcher. The go-to algebraic topology
reference for the non-algebraic topologist.
It is freely available on his website (and linked above).
- The standard reference for virtually anything in PL topology is
Rourke & Sanderson's Introduction to
More accessible (also less general and less formal) is M.A. Armstrong's
Topology, which also has lots of other good stuff.
These papers/book sections are "reasonably" approachable
(to varying degrees).
- For the classification of topological 1-manifolds:
of 1-Manifolds: a Take-Home Exam. David Gale, The American
- For the classification of smooth (compact) 1-manifolds: Appendix 2
of Guillemin & Pollack.
- For existence and uniqueness of smooth structures on surfaces:
The Kirby torus trick for
surfaces. Allen Hatcher, arXiv:1312.3518.
- The first "exotic'' smooth structures, on the 7-sphere:
On Manifolds Homeomorphic to
the 7-sphere. J. Milnor, Annals of Mathematics, 1956.
- There is a nice explanation of the Jordan separation theorem, the Jordan
curve theorem, and the Schoenflies
theorem in Munkres' Topology.
For an algebraic topology proof of the higher-dimensional Jordan
theorem, and an elementary
construction of the Alexander horned sphere, see Appendix 2.B of
- A proof of the
generalized Schoenflies theorem, M. Brown. Bulletin of the AMS,
The title says it all. I have not read this proof but it is amazingly
short (3 pp) and self-contained (1 reference).
- Boy's surface
(at Wikipedia): an immersion of RP^2, the real projective plane, into R^3.
Lectures on Diffeomorphism Groups of Manifolds. Alexander Kupers.
There's a whole lot here, with heavy machinery, but the beginning is nicely
written and approachable.
- Groups of Homotopy Spheres: I.
M. Kervaire and J. Milnor, Annals of Mathematics, 1962.
Describes a group structure (with operation connected sum) on the set of
exotic spheres up to
diffeomorphism in dimensions at least 5 (after the
- The math review
(subscription needed) of J. Cerf's proof (in a series of
four substantial papers written
in French) that "\Gamma_4=0", ie. that
any two OP diffeomorphisms of S^3 are smoothly isotopic.
- Stable homeomorphisms and
the annulus conjecture. R. Kirby, Annals of Mathematics, 1969.
A proof of the Stable
Homeomorphism Conjecture, hence also the Annulus Conjecture, in dimensions
at least 5. This implies by standard arguments
that any two OP homeomorphisms of S^n are isotopic.
Extending Diffeomorphisms, R. Palais. Proceedings of the AMS, 1960.
Theorem B here implies that the connected sum of smooth manifolds
is well defined. The corresponding
result for topological manifolds
uses the Annulus Conjecture.
Hauptvermutung Book, ed. A.A.Ranicki.
Actually a collection of papers disproving the "hauptvermutung der
kombinatorischen topologie", on
homeomorphic simplicial complexes being
isomorphic after subdivision. Ranicki collected them and
wrote a nice
introduction stating the problem and giving context.