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Liam Rivera
Liam Rivera

E Spanier Algebraic Topology Pdf 11


2) Consolidate your mathematical background by working on some relevant classical textbooks first (Kelley's General topology, Dummit&Foote's abstract algebra, Ahlfor's complex analysis, etc). It is not really necessarily for you to learn graduate level algebraic topology at your current mathematics level. It might be condescending for me to suggest this but I believe it is better to read easier stuff than struggle with texts "impossible" for you. The above books are not closely relevant but may be helpful to prepare you to read Hatcher. Also If I remember correctly Hatcher does provide a recommended textbook list in his webpage as well as point set topology notes .




E Spanier Algebraic Topology Pdf 11



3) In case you decide you must learn some algebraic topology, and favor "short" books. You may try this book: introduction to algebraic topology by V.A. Vassilev. This is only about 150 pages but is difficult to read (for me when I was in Moscow). It seems to be available in here. Vassilev is a renowned algebraic topologist and you may learn a lot from that book.


I don't see why I should not recommend my own book Topology and Groupoids (T&G) as a text on general topology from a geometric viewpoint and on 1-dimensional homotopy theory from the modern view of groupoids. This allows for a form of the van Kampen theorem with many base points, chosen according to the geometry of the situation, from which one can deduce the fundamental group of the circle, a gap in traditional accounts; also I feel it makes the theory of covering spaces easier to follow since a covering map of spaces is modelled by a covering morphism of groupoids. Also useful is the notion of fibration of groupoids. A further bonus is that there is a theorem on the fundamental groupoid of an orbit space by a discontinuous action of a group, not to be found in any other text, except a 2016 Bourbaki volume in French on "Topologie Algebrique": and that gives no example applications.


BUT, another part of algebraic topology is in the new jointly authored book Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids (NAT) published in 2011 by the European Mathematical Society. The print version is not cheap, but seems to me good value for 703 pages, and a pdf is available on my web page for the book. Motivation for the methods are given by a thorough presentation of the history and intuitions, and the book should be seen as a sequel to "Topology and Groupoids", to which it refers often.


Part I, up to p. 204, is almost entirely on dimension 1 and 2, with lots of figures. You'll find little, if any, of the results on crossed modules in other algebraic topology texts. You will find relevant presentations on my preprint page.


If you are taking a first course on Algebraic Topology. John Lee's book Introduction to Topological Manifolds might be a good reference. It contains sufficient materials that build up the necessary backgrounds in general topology, CW complexes, free groups, free products, etc.


There is a really well-written but lesser known book by William Fulton. That's the book I learnt Algebraic Topology from. The chapters are laid out in an order that justifies the need for algebraic machinery in topology. A guiding principle of the text is that algebraic machinery must be introduced only as needed, and the topology is more important than the algebraic methods. This is exactly how the student mind works.The book does a great job, going from the known to the unknown: in the first chapter, winding number is introduced using path integrals. Then winding number is explored in a lot more detail, and its connection to homotopy is discussed, without even mentioning fundamental groups. Then a number of results like the Fundamental Theorem of Algebra, Borsuk Ulam and Brouwer's Fixed Point Theorem are proved using winding numbers.Only in Ch.5 do we see the first algebraic object. Here again, the order is flipped: the first De Rham Cohomology group is introduced and used to prove the Jordan Curve Theorem. Then homology groups of open sets in the plane are discussed, and the connection between homology and winding number is made clear. A number of applications to complex integration etc are discussed, and the Mayer-Vietoris theorem is proved for n=1.Covering spaces and fundamental groups are introduced after homology, another novelty. Higher dimensions are encountered only towards the end of the book, but by the time we get there, we already know the general idea behind all the concepts.Very few books take this point of view of developing intuition clarity before generalising rapidly. I think this is really helpful because before studying the general theory of anything, we need to know what it is we are trying to generalise.


You will take pleasure in reading Spanier's Algebraic topology. It is basically "algebraic topology done right", and Hatcher's book is basically Spanier light. Hatcher also doesn't treat very essential things such as the acyclic model theorem, the Eilenberg-Zilber theorem, etc., and he is very often imprecise (even in his definition of $\partial$). There is also no treatment of the very crucial spectral sequences method.


I believe that it is very important to think deeply about whether it is a book, the subject matter, or you that makes a book uneasy to read. we have to confess that algebraic topology is a tough subject. it is nothing like any undergraduate course one takes.


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