Algebraic Topology

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Content: Algebraic topology is concerned with the construction of algebraic invariants (usually groups) associated to topological spaces which serve to distinguish between them. Most of these invariants are ``homotopy'' invariants. In essence, this means that they do not change under continuous deformation of the space and homotopy is a precise way of formulating the idea of continuous deformation. This module will concentrate on constructing the most basic family of such invariants, homology groups, and the applications of these homology groups. I think the organization of the material could be improved. I would move most of chapter 0 to an appendix, as many the unsuspecting undergrad has tried to read that whole chapter before the rest of the book (which I would NOT advise, read it as you need it as much of the motivation comes later). I would also move the category theory material to an appendix. 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. elsewhere, such as the full story on the stable J homomorphism. What is posted now is Version 2.2, dated November 2017. This is a minor revision Ask questions in here or else where (like "ask a topologist") on the problems or sections you found difficult?

notice. The electronic version has narrower margins than the print version for a better reading experience on portable electronic devices. To restore the wider margins for printing a paper copy you can print at 85-90% of full size.The starting point will be simplicial complexes and simplicial homology. An n-simplex is the n-dimensional generalisation of a triangle in the plane. A simplicial complex is a topological space which can be decomposed as a union of simplices. The simplicial homology depends on the way these simplices fit together to form the given space. Roughly speaking, it measures the number of p-dimensional "holes'' in the simplicial complex. For example, a hollow 2-sphere has one 2-dimensional hole, and no 1-dimensional holes. A hollow torus has one 2-dimensional hole and two 1-dimensional holes. Singular homology is the generalisation of simplicial homology to arbitrary topological spaces. The key idea is to replace a simplex in a simplicial complex by a continuous map from a standard simplex into the topological space. It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition. One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes. This result means that we can combine the theoretical power of singular homology and the computability of simplicial homology to get many applications. These applications will include the Brouwer fixed point theorem, the Lefschetz fixed point theorem and applications to the study of vector fields on spheres. As others have said, the book is quite hand wavy. I understand why you wouldn't want to show all the details when you're trying to squeeze *so much stuff* in but PLEASE can I have just a few more details. Nathalie Wahl). Duke Math. J. 155 (2010), 205-269. Here is a pdf file of the version from October 2009 corrections as they come to light. I extend my sincere thanks to all the people who have sent me corrections. Aims: To introduce homology groups for simplicial complexes; to extend these to the singular homology groups of topological spaces; to prove the topological and homotopy invariance of homology; to give applications to some classical topological problems.

This book is seen as the gold standard for a first book on algebraic topology, and I can see why. It has a huge amount of interesting examples, exercises, and pictures, and covers a wide range of topics. The prose, while annoyingly informal at times, helps give an intuition for how mathematicians really think about this stuff, beyond the formalities. The identification diagrams are not quotients of a delta complex, but rather delta complex structures on the quotient space for the square itself. Delta complexes don't behave particularly well under taking quotients, which is what I believe you are observing.Tethers and homology stability for surfaces" (with Karen Vogtmann). Alg. & Geom. Topology 17 (2017), 1871-1916. pdf file.