Math 282B - Homotopy Theory - Winter 2010

Homepage http://math.stanford.edu/~galatius/282B10/

Classes MWF 1:15 - 2:05 in 381-T

Instructor Soren Galatius

Office 383-DD

e-mail galatius@stanford.edu

Office hours By appointment

Prerequisites Math 215B or similar.

Lecture plan. Will be updated during the quarter.

Date Topic

1/4 Introduction: spaces, maps and homotopies. Homotopy groups and their basic properties. The Eckmann-Hilton argument: If a set M is equipped with two different monoid structures, which are compatible (meaning one multiplication map is homomorphism with respect to the other multiplication), and have the same unit, then the two multiplications are equal and commutative.

1/6 Long exact sequence of homotopy groups. [Hatcher 4.3, Bredon chapter VII.5] Homotopy extension property. Cofibrations. [Bredon chapter VII.1, Hatcher page 14 and 460.] Basic example: inclusion of a subcomplex of a CW-complex. n-connected spaces. n-connected maps (warning: Wikipedia's definition is currently wrong).

1/8 Lifting and extension problems. The fundamental theorem was the following: Given a CW pair (K,L) where cells of K-L have dimension at most n, an n-connected map f: X -> Y, h: L -> X, and a map g: K -> Y whose restriction to L is fh, then after possibly changing g by a homotopy relative to L, there exists a map g': K -> X which extends h, and has fg' = g (i.e. g' is the diagonal in the given commutative square, and the two resulting triangular diagrams commute). [Hatcher 4.6, Bredon VII.11.12] Corollaries: If f is n-connected and K is a CW complex, then [K,X] -> [K,Y] is a surjection for dim(K) = n and a bijection for dim(K) < n. A map of CW complexes is a homotopy equivalence if and only if it is a weak equivalence. [Hatcher 4.5, Bredon VII.11.14] If the inclusion A -> X of a subcomplex of a CW complex is a weak equivalence, then it is in fact a strong deformation retract (i.e. the identity map of X is homotopic to a retraction X -> A relative to A). [Hatcher 4.5.] The next topic in the class is fibrations. Homotopy lifting property (HLP), Serre fibrations, Hurewicz fibrations. A Serre fibration has HLP with respect to all CW complexes; in fact it has the relative HLP with respect to all CW pairs. [Hatcher starting on page 405, Bredon chapter VII.6.]

1/11 Note: "fibration" means Serre in Bredon and Hurewicz in Hatcher. We proved: If (K,L) is a CW pair such that the inclusion is a weak equivalence (which we saw last time is the same as L being a deformation retract of K), p: E -> B is a Serre fibration, and there are given maps f: K -> B and g: L -> E with pg = f|L (this can be expressed as a commutative square), then there exists a map K -> E extending g and lifting f (this is then a diagonal map in the square, making the two resulting triangles commute). [Hatcher doesn't explicitly write this result, but it's VII.6.4 in Bredon's book.] Using this, we established the long exact sequence in homotopy associated to a fibration; viz. we proved that a Serre fibration p: E -> B with fiber F = p^-1(b_0) induces an isomorphism from pi_n(E,F,e_0) to pi_n(B,b_0). [Hatcher thm 4.41, Bredon thm VII.6.6.] The proof illustrated a common phenomenon in homotopy theory: proving surjectivity amounts to a lifting problem; injectivity is then a relative version of the same lifting problem. As an example of the long exact sequence we proved that pi_n(CP^oo)=Z for n=2, and 0 otherwise. Finally, we proved that a map p: E -> B being a Serre fibration can be checked locally in B. [Hatcher prop 4.48, Bredon VII.6.11.] (This last property is not quite true for Hurewicz fibrations.)

1/13 Pullback of fibrations. Compact-open topology on Map(X,Y). Remarks about compactly generated spaces. A cofibration A -> X induces a fibration Map(X,Y) -> Map(A,Y). Important example: the path-loop fibration. Fiber homotopy and fiber homotopy equivalence. Homotopy invariance of pullback: If p:E -> B is a fibration and f₀ and f₁ are homotopic maps X -> B, then the two fibrations over X obtained by pullback are fiber homotopy equivalent.

1/15 Pullback of fibrations (a.k.a. fiber product in the category of spaces). Homotopy invariance of pullback: Homotopic maps give fiber homotopy equivalent pullbacks. Corollary: If the base is path connected, then the fibers have the same homotopy type.

1/18 MLK day, no classes.

1/20 Lecture given by Gunnar Carlsson. Homotopy addition theorem. Freudenthal suspension theorem. Stable homotopy groups.

1/22 Lecture canceled.

1/25 Killing homotopy groups (by attaching cells). Eilenberg-MacLane spaces. Postnikov towers. Relative version: Given A -> X, and a natural number n, there is a space Z, obtained from A by attaching cells of dimension greater than n, and a map Z -> X, such that pi_i(Z) -> pi_i(X) is injective for i = n and is bijective for i > n. [Hatcher prop. 4.13. I also explained that the construction can be made functorial by choosing all possible maps].

1/27 CW approximation: If (X,A) is n-connected, then there exists a weak equivalence (Z,A) -> (X,A) where Z is obtained from A by attaching cells of dimension > n. [Hatcher prop. 4.13.] Weak equivalences induce isomorphisms in homology. (We actually proved that if pi_n(X,A) = 0 for all n and all basepoints, then the singular complex C_*(X,A) has a chain contraction.) [Hatcher gives a different proof of this in prop 4.21. The proof I gave in class is similar to Bredon's theorem VII.10.5.] Stated Hurewicz' theorem [Hatcher 4.32]: If (X,Z) is (n-1)-connected with n > 1, and A simply-connected, then pi_n(X,A) is isomorphic to H_n(X,A). Then I explained how to deduce from this, that a map between simply connected spaces is a weak equivalence if and only if it induces an isomorphism in homology groups. (The assumption on simple-connectivity is essential; without it, a famous counterexample is given by the Barratt-Priddy-Quillen theorem). Finally I proved that the natural map pi_n(X,A) -> pi_n(X/A) is an isomorphism under suitable connectivity assumptions [Hatcher prop 4.28]

1/29 Proof of Hurewicz' theorem. Eilenberg-MacLane spaces represent cohomology (statement). Spectra and Omega-spectra.

2/1 Uniqueness of Eilenberg-MacLane spaces. Eilenberg-MacLane spectra. Axioms for cohomology theories. Any Omega-spectrum gives rise to a cohomology theory.

2/3 Eilenberg-MacLane spaces represent cohomology. As a corollary, cohomology of Eilenberg-MacLane spaces calculate natural transformations.

2/5 Converting a map to a fibration. The homotopy fiber of f: X -> Y is the fiber after it is converted to a fibration. If F is the homotopy fiber, then the homotopy fiber of F -> X is homotopy equivalent to \Omega Y. Thus f: X -> Y extends infinitely to the left to a sequence of spaces and maps. Postnikov towers.

2/8 Postnikov towers of principal fibrations. I proved that they exist when X is simply connected. Essentially the same proof gives existence when pi_1(X) acts trivially on pi_n(X) for all n>1.

2/10 Introduction to spectral sequences. We will follow the notes on Hatcher's homepage somewhat. Example: the Serre spectral sequence (theorem 1.3). To construct spectral sequences, we will use exact couples. Definition of exact couple and the derived exact couple.

2/12 Explicit (non-iterative) formula for the (r-1)st derived couple of an exact couple. The exact couple and spectral sequence of a filtered space. General definition of E^\infty (using that a subquotient of a subquotient is a subquotient). Formula for E^\infty for an exact couple. The E^\infty term of the spectral sequence of a filtered space (assuming the filtration starts with the empty space, and that X is the direct limit of the spaces in the filtration). Here are some lecture notes (to be expanded).

2/15 Presidents' day, no classes.

2/17 Construction of the Serre spectral sequence. lecture notes.

2/19 Construction of the Serre spectral sequence. Examples. Preliminary version of lecture notes (updated after class).

2/22 The Serre spectral sequence for an arbitrary Serre fibration. Functoriality. Examples: homology of Omega S^n, Thom isomorphism, Euler characteristic of the total space of a fibration (it's the product of the Euler characteristics of the fiber and the base).

2/24 Products in spectral sequences. Cohomology spectral sequence. Examples. Notes.

2/26

3/1

3/3

3/5

3/8 Lecture given by Mikael Vejdemo Johansson

3/10 Lecture given by Mikael Vejdemo Johansson.

3/12 Last day of class

Date	Topic
1/4	Introduction: spaces, maps and homotopies. Homotopy groups and their basic properties. The Eckmann-Hilton argument: If a set M is equipped with two different monoid structures, which are compatible (meaning one multiplication map is homomorphism with respect to the other multiplication), and have the same unit, then the two multiplications are equal and commutative.
1/6	Long exact sequence of homotopy groups. [Hatcher 4.3, Bredon chapter VII.5] Homotopy extension property. Cofibrations. [Bredon chapter VII.1, Hatcher page 14 and 460.] Basic example: inclusion of a subcomplex of a CW-complex. n-connected spaces. n-connected maps (warning: Wikipedia's definition is currently wrong).
1/8	Lifting and extension problems. The fundamental theorem was the following: Given a CW pair (K,L) where cells of K-L have dimension at most n, an n-connected map f: X -> Y, h: L -> X, and a map g: K -> Y whose restriction to L is fh, then after possibly changing g by a homotopy relative to L, there exists a map g': K -> X which extends h, and has fg' = g (i.e. g' is the diagonal in the given commutative square, and the two resulting triangular diagrams commute). [Hatcher 4.6, Bredon VII.11.12] Corollaries: If f is n-connected and K is a CW complex, then [K,X] -> [K,Y] is a surjection for dim(K) = n and a bijection for dim(K) < n. A map of CW complexes is a homotopy equivalence if and only if it is a weak equivalence. [Hatcher 4.5, Bredon VII.11.14] If the inclusion A -> X of a subcomplex of a CW complex is a weak equivalence, then it is in fact a strong deformation retract (i.e. the identity map of X is homotopic to a retraction X -> A relative to A). [Hatcher 4.5.] The next topic in the class is fibrations. Homotopy lifting property (HLP), Serre fibrations, Hurewicz fibrations. A Serre fibration has HLP with respect to all CW complexes; in fact it has the relative HLP with respect to all CW pairs. [Hatcher starting on page 405, Bredon chapter VII.6.]
1/11	Note: "fibration" means Serre in Bredon and Hurewicz in Hatcher. We proved: If (K,L) is a CW pair such that the inclusion is a weak equivalence (which we saw last time is the same as L being a deformation retract of K), p: E -> B is a Serre fibration, and there are given maps f: K -> B and g: L -> E with pg = f\|L (this can be expressed as a commutative square), then there exists a map K -> E extending g and lifting f (this is then a diagonal map in the square, making the two resulting triangles commute). [Hatcher doesn't explicitly write this result, but it's VII.6.4 in Bredon's book.] Using this, we established the long exact sequence in homotopy associated to a fibration; viz. we proved that a Serre fibration p: E -> B with fiber F = p^-1(b_0) induces an isomorphism from pi_n(E,F,e_0) to pi_n(B,b_0). [Hatcher thm 4.41, Bredon thm VII.6.6.] The proof illustrated a common phenomenon in homotopy theory: proving surjectivity amounts to a lifting problem; injectivity is then a relative version of the same lifting problem. As an example of the long exact sequence we proved that pi_n(CP^oo)=Z for n=2, and 0 otherwise. Finally, we proved that a map p: E -> B being a Serre fibration can be checked locally in B. [Hatcher prop 4.48, Bredon VII.6.11.] (This last property is not quite true for Hurewicz fibrations.)
1/13	Pullback of fibrations. Compact-open topology on Map(X,Y). Remarks about compactly generated spaces. A cofibration A -> X induces a fibration Map(X,Y) -> Map(A,Y). Important example: the path-loop fibration. Fiber homotopy and fiber homotopy equivalence. Homotopy invariance of pullback: If p:E -> B is a fibration and f₀ and f₁ are homotopic maps X -> B, then the two fibrations over X obtained by pullback are fiber homotopy equivalent.
1/15	Pullback of fibrations (a.k.a. fiber product in the category of spaces). Homotopy invariance of pullback: Homotopic maps give fiber homotopy equivalent pullbacks. Corollary: If the base is path connected, then the fibers have the same homotopy type.
1/18	MLK day, no classes.
1/20	Lecture given by Gunnar Carlsson. Homotopy addition theorem. Freudenthal suspension theorem. Stable homotopy groups.
1/22	Lecture canceled.
1/25	Killing homotopy groups (by attaching cells). Eilenberg-MacLane spaces. Postnikov towers. Relative version: Given A -> X, and a natural number n, there is a space Z, obtained from A by attaching cells of dimension greater than n, and a map Z -> X, such that pi_i(Z) -> pi_i(X) is injective for i = n and is bijective for i > n. [Hatcher prop. 4.13. I also explained that the construction can be made functorial by choosing all possible maps].
1/27	CW approximation: If (X,A) is n-connected, then there exists a weak equivalence (Z,A) -> (X,A) where Z is obtained from A by attaching cells of dimension > n. [Hatcher prop. 4.13.] Weak equivalences induce isomorphisms in homology. (We actually proved that if pi_n(X,A) = 0 for all n and all basepoints, then the singular complex C_*(X,A) has a chain contraction.) [Hatcher gives a different proof of this in prop 4.21. The proof I gave in class is similar to Bredon's theorem VII.10.5.] Stated Hurewicz' theorem [Hatcher 4.32]: If (X,Z) is (n-1)-connected with n > 1, and A simply-connected, then pi_n(X,A) is isomorphic to H_n(X,A). Then I explained how to deduce from this, that a map between simply connected spaces is a weak equivalence if and only if it induces an isomorphism in homology groups. (The assumption on simple-connectivity is essential; without it, a famous counterexample is given by the Barratt-Priddy-Quillen theorem). Finally I proved that the natural map pi_n(X,A) -> pi_n(X/A) is an isomorphism under suitable connectivity assumptions [Hatcher prop 4.28]
1/29	Proof of Hurewicz' theorem. Eilenberg-MacLane spaces represent cohomology (statement). Spectra and Omega-spectra.
2/1	Uniqueness of Eilenberg-MacLane spaces. Eilenberg-MacLane spectra. Axioms for cohomology theories. Any Omega-spectrum gives rise to a cohomology theory.
2/3	Eilenberg-MacLane spaces represent cohomology. As a corollary, cohomology of Eilenberg-MacLane spaces calculate natural transformations.
2/5	Converting a map to a fibration. The homotopy fiber of f: X -> Y is the fiber after it is converted to a fibration. If F is the homotopy fiber, then the homotopy fiber of F -> X is homotopy equivalent to \Omega Y. Thus f: X -> Y extends infinitely to the left to a sequence of spaces and maps. Postnikov towers.
2/8	Postnikov towers of principal fibrations. I proved that they exist when X is simply connected. Essentially the same proof gives existence when pi_1(X) acts trivially on pi_n(X) for all n>1.
2/10	Introduction to spectral sequences. We will follow the notes on Hatcher's homepage somewhat. Example: the Serre spectral sequence (theorem 1.3). To construct spectral sequences, we will use exact couples. Definition of exact couple and the derived exact couple.
2/12	Explicit (non-iterative) formula for the (r-1)st derived couple of an exact couple. The exact couple and spectral sequence of a filtered space. General definition of E^\infty (using that a subquotient of a subquotient is a subquotient). Formula for E^\infty for an exact couple. The E^\infty term of the spectral sequence of a filtered space (assuming the filtration starts with the empty space, and that X is the direct limit of the spaces in the filtration). Here are some lecture notes (to be expanded).
2/15	Presidents' day, no classes.
2/17	Construction of the Serre spectral sequence. lecture notes.
2/19	Construction of the Serre spectral sequence. Examples. Preliminary version of lecture notes (updated after class).
2/22	The Serre spectral sequence for an arbitrary Serre fibration. Functoriality. Examples: homology of Omega S^n, Thom isomorphism, Euler characteristic of the total space of a fibration (it's the product of the Euler characteristics of the fiber and the base).
2/24	Products in spectral sequences. Cohomology spectral sequence. Examples. Notes.
2/26
3/1
3/3
3/5
3/8	Lecture given by Mikael Vejdemo Johansson
3/10	Lecture given by Mikael Vejdemo Johansson.
3/12	Last day of class