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The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.
Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , which records information about loops in a space.
Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains , cocycles , and coboundaries.
Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory. A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane , the sphere , and the torus , which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions.
Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points , line segments , triangles , and their n -dimensional counterparts see illustration.
Homology Theory [electronic resource] : an Introduction to Algebraic Topology
Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. Whitehead to meet the needs of homotopy theory.
This class of spaces is broader and has some better categorical properties than simplicial complexes , but still retains a combinatorial nature that allows for computation often with a much smaller complex.
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An older name for the subject was combinatorial topology , implying an emphasis on how a space X was constructed from simpler ones  the modern standard tool for such construction is the CW complex. In the s and s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups , which led to the change of name to algebraic topology.
In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism or more general homotopy of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statement easier to prove.
Two major ways in which this can be done are through fundamental groups , or more generally homotopy theory , and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of a finite simplicial complex does have a finite presentation.
Vick begins with singular homology, which is far easier to set up on a technical level than its simplicial cousin.
After a number of very worthwhile deductions from the singular theory, he introduces CW complexes and shows that their homology is the same as that obtained from the singular theory. His treatments of products and Poincare duality are also outstanding. The best thing about this book is that it's genuinely accessible to the beginning graduate student.
You don't need to have had a course in category theory to read and understand this book. In fact, seeing some things in a concrete setting here, like direct limits, will undoubtedly help people understand their more abstract categorical versions later on in graduate school. I will make one criticism of this book's way into homology theory, and that is that Vick neglects to prove Hurewicz's result that the first homology group of a path-connected space is the abelianization of the fundamental group. It also helps in solving one of the exercises in this book, where one has to construct a map from the n-sphere to itself with any given integer as its degree; once you understand the Hurewicz result, it's easy to find such a map on the circle in the complex plane, and then repeatedly iterate the suspension operator to increase the dimension of the sphere.
Apart from this detail, I love everything about this book. There are some typos, but that's true in most books, and they didn't bother me that much. I recommend this book to any novice student of algebraic topology, and especially to any who feel that their background in abstract algebraic machinery category theory, homological algebra is somewhat lacking.
You'll learn it well from this book. Enjoy it. This introduction to singular homology combines a strong historical sense with an easy mastery of modern methods.
The massive contributions of Poincare and Brouwer are credited, and their geometrical motivations are clear. At the same time the book neither minimizes nor apologizes for modern algebraic machinery, but treats categories and acyclic models and more as natural means to simplify the subject.
The book goes through Poincare duality and a good account of the Lefschetz fixed point theorems. It is at once very visual and algebraically slick. The only problem with this approach is that the author seems a bit uncomfortable descending into the nuts and bolts of the longer proofs of two key results the acyclic model theorem, and the duality theorem.
He handles the details unevenly and makes some actual mis-statements. Here the reader needs the experience and confidence to make some corections. This was the textbook for the first third of a year-long algebraic topology sequence at Oregon State in We were told by the prof that Vick was a student of Stong and that the book was essentially Stong's course written up with his blessing. It's hard to ask for a better pedigree than that, as Stong was a legend for his teaching as well as his research.go site
Homology theory, Volume 53 - 1st Edition
Although there are some minor quibbles noted in the 4-star reviews , I still haven't found a better treatment of the key results, nor a more direct path. The proof of Poincare duality in particular is that of Hans Samelson, another legend in the field for both teaching and research. Checking the references, one finds that this was not the only such example where Vick sought out what was then regarded as the best proof available for beginners.
It is also noteworthy that community consensus on which are best has not changed much, if any, since then. The plethora of typos may be a "feature" of the reprint, since I don't recall that many in the original Acad. Press edition we used, and I still have. As should be clear, this one is a real keeper.
Related Homology Theory: An Introduction to Algebraic Topology
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