Javascript is not enabled in your browser. Enabling JavaScript in your browser will allow you to experience all the features of our site. Learn how to enable JavaScript on your browser. This self-contained text is suitable for advanced undergraduate and graduate students and may be used either after or concurrently with courses in general topology and algebra. It surveys several algebraic invariants: the fundamental group, singular and Cech homology groups, and a variety of cohomology groups.

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Proceeding from the view of topology as a form of geometry, Wallace emphasizes geometrical motivations and interpretations. Once beyond the singular homology groups, however, the author advances an understanding of the subject's algebraic patterns, leaving geometry aside in order to study these patterns as pure algebra.

Numerous exercises appear throughout the text. In addition to developing students' thinking in terms of algebraic topology, the exercises also unify the text, since many of them feature results that appear in later expositions. Extensive appendixes offer helpful reviews of background material.

Wallace is Professor Emeritus of Mathematics at the University of Pennsylvania and the author of two other Dover books.

Table of Contents Preface 1. Singular Homology Theory 2. Singular and Simplicial Homology 3. Chain Complexes—Homology and Cohomology 4. The Cohomology Ring 5. Cech Homology Theory—The Construction 6. Further Properties of Cech Homology Theory 7. Cech Cohomology Theory Appendix A.

## Modern Applications of Homology and Cohomology

The Fundamental Group Appendix B. See All Customer Reviews. Shop Books. Add to Wishlist.

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USD Hardcover FREE. Sign in to Purchase Instantly. Overview This self-contained text is suitable for advanced undergraduate and graduate students and may be used either after or concurrently with courses in general topology and algebra. About the Author Andrew H. General Topology Bibliography Index. Average Review. MathOverflow is a question and answer site for professional mathematicians.

It only takes a minute to sign up. In intuitive terms, what is the main difference? This is, more or less, the number of holes in the complex. But what is the geometrical interpretation of cohomology? As to what cohomology actually measures, I think a general theme is "the failure of locally trivial things to be globally trivial", or perhaps "the failure of local solutions to glue together to form a global solution". Let me give a somewhat elementary answer. Like a lot of things, this stuff presumably came out of calculus.

This can be carried out in higher dimensions, as well. At first glance cohomology seems completely dual to homology, and therefore seemingly redundant. But in fact it has more structure. Since you multiply wedge differential forms together, cohomology becomes a ring.

This is still true in more general approaches such as singular cohomology. On the homology side, one has an intersection pairing, but this is harder to describe and only available for really "nice" spaces. Perhaps another feature of cohomology worth mentioning is that is contravariant: cohomology classes pullback from the target to the source under a map of spaces. This important in the theory of characteristic classes, where such classes are pulled back from to maps to certain universal spaces.

Such classes measure the amount of "twisting" of bundles. On a closed, oriented manifold, homology and cohomology are represented by similar objects, but their variance is different and there is an important change in degrees.

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For simplicity, consider homology or cohomology classes represented by submanifolds. That's one reason I think that cohomology is of more use in algebraic geometry. This point of view is more applicable than it might seem since in a manifold with boundary cohomology classes are similarly defined by submanifolds whose boundary lies on the boundary of the ambient manifold.

Since any finite CW complex is homotopy equivalent to a manifold with boundary, one can view cohomology in this way for finite CW complexes and often infinite ones as well. Cohomology is a graded ring functor, homology is just a graded group functor. As groups cohomology does not give anything that homology does not already provide.

Whatever geometric interpretation you have for homology would mostly probably work also for cohomology. But the multiplication in cohomology allows better differentiation between topological spaces which is not possible with homology. In this sense cohomology is a finer invariant. Specific examples can be found in the book of Spanier. There are extraordinary cohomology theories, cobordism, K-theory, etc.

These satisfy most of the Eilenberg-Steenrod axioms.

### Learning Outcomes

Also cohomology can be generalized to algebraic geometry, which is very important. One cannot stress how important this is. Cohomology is the king there. One helpful way of thinking of integral cohomology maybe the following.

## Algebraic Topology (Master)

In homology, you look at sums of simplices in the topological space, upto boundaries. In cohomology, you have the dual scenario, ie you attach an integer to every simplex in the topological space, and make identifications upto coboundaries. I think Hatcher's book has a good elementary exposition of some of the differences between homology and cohomology.