Defining homotopy and homotopy groups is fairly easy, but their actual calculation or determination is very difficult in general. This book, written by one of the major contributors of the subject, is comprehensive and can serve as both a textbook and as a reference since there are collections of exercises at the end of each chapter. The author also includes some brief historical background from time to time, making the reading much more palatable. Space forbids a detailed review, so only the main points of each chapter will be discussed.
Chapter 1: Elementary notions of homotopy are discussed, with emphasis on map extension and maps of the n-sphere into itself. Compactly generated spaces are introduced and all the spaces in the book are assumed to be compactly generated. The homotopy lifting property is used to motivate the concept of a fibration. Unfortunately the Hurewicz theorem on fibrations over paracompact spaces is not proven. The proof involves partitions of unity and is very illustrative of typical methods. Best part of the chapter: the discussion on the duality between a fibration and a cofibration.
Chapter 2: This is a review of CW-complexes, which are introduced as the easiest objects to study in algebraic topology. The author covers the homology and cohomology theory of CW-complexes cellular maps. The best part of the chapter though is on products and the cohomology ring: the author explains in detail and gives special insight as to why calculation of cup products is difficult for a general CW-complex (there is no Alexander-Cech-Whitney formula for doing this, as is the case for simplicial complexes).
Chapter 3: The author considers the task of putting a natural product on the set of homotopy classes of mappings. This results in a discussion of H- and H'-spaces and eventually their homology. The fundamental group makes its appearance here, along with Hopf algebras, the latter being very important recently in the context of "quantum groups". Best part of the chapter: the discussion on the need for the suspension and smash products to alleviate the dependence on base points.
Chapter 4: The author concentrates on homotopy groups in more detail, asking first to what extent homotopy can be viewed as having the same properties as homology. Defining first relative homotopy groups and the higher homotopy groups, the connection between homotopy and homology is eventually done via the Hurewicz map and the Whitehead theorem. The best part of the chapter is on the difference between homology groups and homotopy groups under cofibrations, with the opposite occurring for fibrations.
Chapter 5: The author returns to the study of CW-complexes, where he studies their (relative) homotopy groups. The relative homotopy group for n greater than or equal to 3 is calculated and shown to be a free Z-module over the first homotopy group of the subcomplex with one basis element for each n-cell, in analogy to the homology of CW-complexes, wherein the nth homology group is free abelian with one basis element for each n-cell of the pair. Eilenberg-Maclane spaces are introduced here for the first time. By far the best part of the chapter though is the treatment of obstruction theory.
Chapter 6: Because of its importance to the theory of characteristic classes, and because it is a nice overview of what the topologist Norman Steenrod began in his book on fiber bundles, everything in this chapter is interesting. It is shown in detail how to assign to every pair a homology group with local coefficients. Since characteristic classes are so very important in topology, and since their treatment in the literature is usually too formal, this chapter is a real treat, as it offers rare insight into the origins and intuition behind characteristic classes.
Chapter 7: The homology groups of a fibration are discussed, with attention initially made to those whose base space is a suspension of another space. This motivates the James reduced products for the loop space of a suspension. The best part though is the detailed treatment of fibrations with spherical fiber.
Chapter 8: The homology groups of the loop space of a space B are related to those of B via the homology suspension. This discussion leads to the best part of the chapter: the discussion of stable operations and Steenrod squares.
Chapter 9: Eilenberg-Maclane spaces are used build a space of given homotopy type. The maps from one component to one of lower dimension leads to the Postnikov invariant of the space and then Postnikov systems, which is the best part of the chapter.
Chapter 10: The Lusternik-Schnirelmann category arises here, and used to prove nilpotency of homotopy mappings of a space to a group. The best part is the one on the Whitehead product.
Chapter 11: Homotopy operations are treated in detail, being motivated by the desire to emulate the ability of Eilenberg-MacLane spaces to give universal examples of cohomology operations. The proof of the Hilton-Milnor theorem, which gives a connection between the homotopy groups of the wedge of two spaces and the homotopy of the spaces themselves, is the best part of the chapter.
Chapter 12: Stable homotopy groups are discussed and shown to behave like homology groups, at least if the dimension axiom is relaxed. In that regard the discussion on the comparison with the Eilenberg-Steenrod axioms is the best part of the chapter.
Chapter 13: This is the most difficult of all the chapters in the book, for it introduces the method of spectral sequences for studying the homology of fibrations. Closely connected with later work on K-theory, spectral sequences are give an elegant treatment here, making this chapter also one of the best in the book. The author shows in detail how to associate with the fiber map an exact couple whose spectral sequence connects the homology of the base with local coefficients in the fiber, to the homology of the total space.

I am director of secondary student teachers at Lindenwood University in St. Charles, Missouri. I teach courses in Middle School Philosophy, Educational Psychology, and Teaching and Classroom Management. I have found this book to be very helpful in presenting information about the basics of the middle school philosophy. It encourages educators to seek excellence in the education of youth during this critical time in their development.

This text, initially published at in the first half of this century, is an amazing read. This is ostensibly a book about how fish reproduce and how the desire to procreate leads to incredible behavioral patterns that were only just being understood. Roule's passion about the subject, however, takes the book out of the realm of mere description as he pays tribute to these fascinating animals. The content of the book focuses primarily on the travels of salmon and the European eel, while also touching upon the "migratory" behavior of herring, shad, tunny, trout, sturgeon, mackerel, cod and other fish species. The treatment of salmon and eel is where Roule it his best: he is successful at adequately portraying the magnitude of what these species go through in order to reproduce. The salmon's sojourn from the depths of the ocean to the mountains of the continent, and the eels journey in the opposite direction, from Europe to Bermuda, are, to Roule, focal points in understanding the behavior of fish. All other migrations are simply smaller versions of these great treks. Roule also presciently touches upon those environmental problems that will wreak havoc on some of the world's fisheries shortly after the first publication of the book.

This is one of the best introductions to the subject that I've read. The explanations are transparent with ample illustrations and examples. There are also many exercises to help the reader enforce what he has learned. There is also a very helpful annoted bibliography (actually the Afterword) which guides the reader to more advanced reading.

It is really a shame that this book is not quoted more often. Overall, this is a very fine book.

We don't play soccer, well we almost played field hockey but we didn't make the team. but that doesn't matter, because this book is really about best friends even if they don't play soccer, and we're best friends. Alice and Tiffany are our names and we love this book. because even if you lie sometimes you didn't mean to be bad and your friends can forgive you and God still loves you.

William J. Boerst's Generous Anger is a 112-page biography of George Orwell will appeal to students from elementary grades 6 through middle school: it provides an account of his life, early influences, and the writings which changed his world. Facts read as lively as fiction in this biographical portrait.

If you love maps, Grand Canyon, and geology, you will want to have this marvelous map on your wall. It's a big, dramatically colorful geologic map, covering the entire eastern region of the Canyon, where most of the hiking trails are located. Grand Canyon geology is simple and bold, and therefore eminently accessible to the amateur. This map, which makes it even more accessible, is a wonderful companion piece to the handful of geology books about the Canyon, and will inspire you to purchase one if you have not already done so. With a little effort, using the legend in the margin, you will soon learn to pick out every layer of rock that you can observe from the rim or encounter on your hike or float. Needless to say, the map is even more of a treasure-trove to the professional. More than a map, the beautifully printed sheet also includes several cross-sections of the Canyon, revealing what is going on beneath the surface. If you have visited the South Rim you may have seen this map. Two copies are on display in the entrance hall / cocktail lounge of the Arizona Steak House, adjacent to Bright Angel Lodge. Because of its size, it is not a map you would want to carry in your pack, but, tacked up on your wall or unrolled on a table, it will provide plenty of rock-solid lore as you plan your next Canyon adventure or remember the magic of your last visit.

This is a very easy book to read; I hold this out as a virtue since George Grant was a very complicated man. Christian does well to stick to the facts of Grant's life and then when necessary add insight to the scaffolding. There is a human quality present throughout. The all-too-human is not totally absent; although, it is obvious the author greatly admires Grant, and it is apparent Grant was his ad-hoc tutor, friend and dinner guest on many occasions. The fruits of this familiarity are gossipy tidbits about Grant's dinner with Leo Strauss and Grant's squabbles with Fulton Anderson of the University of Toronto, among other stories. This biography can serve as an introduction to many of Grant's philosophical inspirations- Simone Weil, Martin Heidegger, and Leo Strauss- as well as documenting an important part of Canada's political and social progress (maybe Grant would use a word other than progress) from a colonial state to the fragile position it now holds in the modern world.

This book is excellent first and foremost because it brings together almost all of the pictorial representations of George Washington ever produced. The book includes the artwork of Gilbert Stuart, John Wollaston, Jean Leon Gerome Ferris, Charles Willson Peale, and many others in addition to engravings. This book also contains a vast text which is lengthy by necessity enabling the book to cover the entirety of Washington's life and includes the development of historical thought in the years since Washington's death. The book also includes an enormous amount of interesting information on Martha Custis and her children and other family relations. Whether a person is a devoted researcher of George Washington himself or the colonial period or the American Revolution years or all of the above, this book makes an excellent addition.

When I was a young boy I saw the Jack Paar show? and one day Paar was interviewing a family that had been captured by the Japanese during World War II and were interned in a P.O.W. camp. This family suffered tremendously. The man, William R. Moule, wrote a book and I still remembered the title from 40 years ago "God's Arms Around Us."

Well to make a long story short, I searched Amazon .com for the title 40 years later and they had it!! The book was extremely exciting, I am surprised it has never been made into a movie.

Mr. Moule's writing style was very personal, he has the ability to make you feel you are right there with him, his wife and 3 small children trying to escape the Japanese who are looking for them.

I highly recommend this book to anyone that has any relative who fought in the Second World War or has an interest in WW II. It was really great!!