## Highlights - “classical homotopy theory”\. It contains cellular approximation, the Hurewicz theorem, properties of H-spaces and co-H-spaces, Whitehead and Samelson products, and a large amount of space devoted to the manipulations of fibrations and cofibrations\. It also introduces the James construction, Hopf-invariant maps, and (Paul Selick 13) - Lusternik-Schnirrelmann category\. (Paul Selick 14) - For example, many product and wedge decompositions which result from manipulating fibrations and cofibrations are given\. These are well known and in everyday use but not easy to find in the literature\. (Paul Selick 14) - Chapter 8 contains the basics of simplicial sets\. It discusses Kan complexes, the singular complex, simplicial groups, simplicial abelian groups, and how the homotopy theory of simplicial sets parallels that of topological spaces (Paul Selick 14) - the simplicial classifying construction and simplicial loop construction, is omitted and readers who intend to make large use of simplicial sets will need to consult a book devoted explicitly to that subject (Paul Selick 14) - simplicial sets are used primarily in discussing localization (Paul Selick 14) - Chapter 9 is a brief introduction to fibre bundles (Paul Selick 14) - Basic definitions, classification theorems, and Milnor’s construction are given\. A student of algebraic topology would be well advised to read more on this subject, for example in Milnor’s “Characteristic Classes” [MS]\. (Paul Selick 14) - The material of Chapter 10 on Hopf algebras is taken mostly from the classic paper by Milnor and Moore [MM] (Paul Selick 14) - Chapter 11 is a long chapter on spectral sequences\. (Paul Selick 14) - the major spectral sequences used in homotopy theory are discussed individually in the second half of the chapter\. (Paul Selick 14) - The major spectral sequence in homotopy theory omitted from these notes is the Adams spectral sequence\. (Paul Selick 14) - Readers are referred to the companion volume by Stan Kochman in this monograph series for a complete discussion of the Adams spectral sequence\. (Paul Selick 14) - For the first Eilenberg-Moore spectral sequence we take a very algebraic approach closely related to Eilenberg and Moore’s original proof\. This allows the demonstration of homological techniques useful in other contexts and also demonstrates clearly how much information is lost when one passes from the singular chain complex on a space to its homology\. (Paul Selick 14) - although it would make sense to first develop the properties of a category of spectra to avoid having to express the statements in an awkward form\. See Smith [Sm] for a derivation from this point of view\. (Paul Selick 15) - Chapter 12 discusses localization and completion\. (Paul Selick 15) - The main purpose of Chapter 13 is to develop the connection between generalized cohomology theory and representing spaces, and in particular the Brown Representability Theorem\. (Paul Selick 15) - Spectra and stable homotopy concepts are introduced, (Paul Selick 15) - The reader is also referred to the books by Adams [A2] and Lewis, May, Steinberger [LMS] for detailed construction of categories of spectra and their properties (Paul Selick 15) - nother important universal construction is the “pullback”\. (Paul Selick 28) - The dual concepts, obtained by reversing all of the arrows, are called “coproduct” and “pushout” respectively\. (Paul Selick 28) - A diagram in a category (Paul Selick 29) - Dually a colimit of the diagram D (Paul Selick 29) - limit of the diagram D (Paul Selick 29) - A pullback is the limit of a diagram (Paul Selick 29) - Definition 1\.3\.1 A category C is called an additive category if: (Paul Selick 29) - Definition 1\.3\.2 A category C is called an abelian category if: (Paul Selick 30) - Theorem 2\.3\.3 \(Tychonoff’s Theorem\) If X; is compact for all j € J then [1;c; Xj is compact\. (Paul Selick 33) - Definition 2\.3\.4 A topological space X is called locally compact if every point has a neighbourhood whose closure is compact\. (Paul Selick 33) - Theorem 2\.3\.8 \(Exponential Law\) If X is locally compact Hausdorff then \(YX\)2 2 YZXX for allY and Z\. (Paul Selick 34) - a continuous map f : X — Y is called cellular if f \(X \(n\) C Y® for all n\. (Paul Selick 37) - Theorem 7\.5\.1 \(Cellular Approximation Theorem\) Let f : X — Y\. Then there exists g : X — Y such that g ~ f and g is cellular\. (Paul Selick 38) - Theorem 7\.5\.8 \(CW Approximation Theorem\) Given a topological space Y there exists a CW-complex X and a map f: X — Y such that fu : m\(X\) — mn \(Y\) is an isomorphism for all n\. (Paul Selick 38) - Theorem 2\.7\.3 \(Milnor\) If X is a CW-complex and K is compact, then XX is a CW -complez\. (Paul Selick 38) - The smash product of pointed spaces X, Y, written X AY, is defined as \(X x Y\)/\(X VY\)\. (Paul Selick 39) - Theorem 3\.1\.2 \(Exponential Law\) Let X,Y, and Z be pointed spaces with X locally compact Hausdorff\. Then \(YX\)% x YZ/X, (Paul Selick 39) - If \(X,z0\) = \({zo}, zo\) then \(X, xo\) is called contractible\. (Paul Selick 39) - X and Y are called homotopy equivalent o (Paul Selick 39) - If Y is path connected and 7;\(Y\) = {1} \(where {1} denotes the trivial group\) then Y is called simply connected\. (Paul Selick 40) - fundamental groups need not be abelian\. (Paul Selick 40) - There are two standard methods of computing homotopy groups: coming down from above by means of “covering spaces” and coming up from within by Van Kampen’s Theorem\. (Paul Selick 40) - Remark A groupoid is a category in which every morphism is an isomorphism\. (Paul Selick 41) - covering projection (Paul Selick 41) - a sheet over U\. (Paul Selick 41) - A space together with an action of G on that space is called a G-space\. (Paul Selick 41) - Proposition 3\.2\.3 Suppose a group G acts on a space X such that for all xz € X there exists an open neighbourhood V, such that V, N\(gVz\) = 0 for all g # e in G\. Then the quotient map p: X — X/G is a covering projection\. (Paul Selick 41) - A key feature of covering spaces is the ability to uniquely “lift” maps from X to E\. (Paul Selick 41) - This theorem implies that a covering projection is a fibration\. (Paul Selick 42) - Corollary 3\.2\.8 A simply connected locally path connected covering space is a universal covering space\. (Paul Selick 43) - Theorem 3\.2\.9 Every connected, locally path connected, semilocally simply connected space has a universal covering space\. (Paul Selick 43) - semilocally simply connected (Paul Selick 43) - Theorem 3\.2\.10 Let p : E — X be a covering projection such that E is simply connected and locally path connected \(thus a universal covering space\)\. Then m1\(X\) £2 the group of covering transformations of p\. (Paul Selick 43) - Exercise 3\.2\.14 Show that [];2, S1 \(with the product topology\) does not have a universal covering space\. (Paul Selick 43) - Theorem 3\.2\.15 \(“Galois Theory” of Covering Spaces\) (Paul Selick 44) - 3 Van Kampen’s Theorem (Paul Selick 44) - amalgamated free product (Paul Selick 44) - Theorem 3\.3\.1 \(Van Kampen’s Theorem\) Let U, V be path connected open subsets of X such that UUV = X and UNV is nonempty and path connected\. Then m1\(X\) = m1 \(U\) \*x, wav\) m1\(V\)\. (Paul Selick 44) - Definition 4\.1\.1 A chain complex (Paul Selick 45) - cycles\. (Paul Selick 45) - boundaries\. (Paul Selick 45) - chain map (Paul Selick 45) - the condition that f be a chain map is usually written fd =df\. (Paul Selick 45) - The suspension of the chain complex (Paul Selick 45) - If H,\(C\) = 0 for n # 0 then C is called acyclic\. (Paul Selick 45) - in the\. category of abelian groups (Paul Selick 45) - 0 — A J\.B 2 C — 0 exact = [ is injective, g is surjective, and C= B/f\(A\)\. (Paul Selick 46) - split monomorphism (Paul Selick 46) - retraction\) (Paul Selick 46) - The map 0, called the connecting homomorphism, is defined as follows\. (Paul Selick 46) - Lemma 4\.1\.7 Let 0 — P 2, Q 5 R — 0 be a short exact sequence of chain complexes\. Then there is an induced natural \(long\) exact sequence of homology groups (Paul Selick 47) - A chain homotopy (Paul Selick 47) - chain homotopy equivalent (Paul Selick 47) - Set X] = A, ® X, ® A,—1\. (Paul Selick 48) - The chain complex X’ is called the algebraic mapping cylinder of f and the quotient complex X'/A is called the algebraic mapping cone of f by analogy with their geometric counterparts\. (Paul Selick 48) - Theorem 4\.2\.4 \(Homology Commutes with Direct Limits\) (Paul Selick 48) - Homology does not commute with inverse limits (Paul Selick 49) - An R-module satisfying the conditions of Proposition 4\.3\.1 is called a projective R-module\. (Paul Selick 49) - A nonnegatively graded chain complex P, of R-modules is called a projective resolution of M if P, is a projective (Paul Selick 49) - R-module for all n and if (Paul Selick 50) - A projective resolution P, exists for every R-module M; in fact it is always possible to construct a resolution in which P, is free for all n\. (Paul Selick 50) - A functor which preserves exactness is called exact\. F is exact if and only if F' preserves kernels and cokernels\. A functor which preserves kernels is called left exact and a functor which preservers cokernels is called right exact\. (Paul Selick 50) - A module @ having the property that tensoring with \) preserves exactness \(equivalently, Tor \(Q, N\)=0 for all n > 0 and all N\) is called flat\. (Paul Selick 51) - In particular, if R is a commutative ring then Tor,\*\(M, N\) becomes an R-module\. (Paul Selick 51) - In general M ®gr N has no natural R-module structure\. (Paul Selick 51) - The algebraic group homology of G is defined by H28\(G\) = Tor2l€\)\(z,7\)\. (Paul Selick 52)