Best book for undergraduate study algebraic topology

[Jianphys17]

In your opinion what is the best book for a first approach to algebraic topology, for self studt more properly!

 

[micromass]

Please tell us your current knowledge and your specific goals.

 

[MathematicalPhysicist]

Croom's book seems like a good coverage of basic algebraic topology;

I plan to read from it after I am finished with Munkres Topology textbook.

After these two basic general topology and algebraic topology we have a continuation of Munkres' in Elements of Algebraic Topology, and Massey's textbook including Bott and Tu's and Bredon's books.

But first go through Munkres' and Croom's.

 

[MathematicalPhysicist]

BTW there are plenty of books on CW complexes (I can cite at least two books which solely touch on only CW complexes); and there are also books that solely cover homological algebra which is important topic also in algebraic topology.

 

[Jianphys17]

Of course i have some general topology prerequisites ! I've the J. Rotman "an intro to a.t." and the Hatcher's book in pdf format, which is better?

 

[MathematicalPhysicist]

I tried reading Hatcher's book, and it seems as the standard textbook nowadays, but I didn't like it; its structure seems a bit messy, but if you plan in the future to read more advanced stuff you bound to get distorted by the order of stuff that it's covered.

I also read the first few pages of Croom's book, and it seems better as an undergarduate introduction to basic AT.

I don't have an experience with Rotman's book.

 

[bacte2013]

Ryszard Engelking's Topology - A Geometric Approach has very good exposition on the algebraic and differential topology.

 

[mathwonk]

I suggest as elementary a book as possible; i myself would start with massey for the fundamental group, or artin/braun for homology. bill futon's book is also nice, but i would consider bott and tu as relatively advanced. experts i know use hatcher for beginning college or grad classes, and it is free. milnor's book is also a great classic.https://www.amazon.com/gp/product/0387902716/?tag=pfamazon01-20

http://www.barnesandnoble.com/w/introduction-to-algebraic-topology-emil-artin/1007109174

https://www.amazon.com/dp/0387943277/?tag=pfamazon01-20

https://www.math.cornell.edu/~hatcher/AT/ATpage.html

http://teachingdm.unito.it/paginepersonali/sergio.console/Dispense/Milnor Topology from #681EA.pdf

 

[Jianphys17]

Thanks very much!

 

[Cagin Cevik]

Thank you for sharing.

 

[mathwonk]

I began to appreciate some ideas of algebraic topology originally through differential topology, in the form of my own discovery of the meaning of the advanced calculus theorem called greens theorem. The problem was to comp0ute the fundamental group of the punctured plane, R^2 - (0,0). I.e. we want to show that the unit circle cannot be moved continuously, within the punctured plane, so as to no longer wrap around the origin. We weaken our request to try to show rather that it cannot be so moved differentiably. Take enough time, as much as you need, to realize that moving the unit circle off the origin, since one can then shrink it to a point, means finding a differentiable map from the unit disc to the punctured plane, so that the boundary maps to the unit circle, and the rest of the disc maps into the punctured plane.

So we want to show that no differentiable map of the unit disc to the plane can map the boundary identically to the unit circle, but the rest of the disc misses the oprigin. This follows from the greens theorem. I.e. that theorem says you can compute a certain integral either around the boundary of the disc, or over the disc itself. The second ingredient is the "angle form" dtheta, whose integral is non zero over the unit ciurcle, but whose curl is zero, hence the integral of its curl is zero over the disc. If the map we described were to exist, the greens theorem would then say that zero equals a non zero number, hence impossible.

geometrically one can define the conceopt of "winding number", and use green to prove that winding number does not change under a differntiable motion of the curve. Then one computes that the winding number of the unit circle is one, while that of a circle moved of the origin is zero.

this gets generalized to the concept of fundamental group, or homology of the punctured plane, whihc turns out to be a group whihc is generated by that angle form. I.e. if you understahd the angle form you have understood the fundamental group of the punctured plane as well as the first homology group.

ramping up, there is a solid angle form that generates the second homology group of the punctured 3 space, and let's you prove there are no non zero vector fields on the 2 sphere.

I had taken courses in algebraic topology and understood nothing, and then while teaching calculus i asked muyself what good was greens theorem? the answer i dioscovered unlocked the ideas of differential and algebraic topology for me. probably fulton's book and its approach, is closest to my own path.