SOME EXERCISES. This is not an assignment, though some exercises on this list might become part of an assignment. Class 2

Transcription

1 SOME EXERCISES This is not an assignment, though some exercises on this list might become part of an assignment. Class 2 (1) Let C be a category and let X C. Prove that the assignment Y C(Y, X) is a functor C(, X) : C op Sets. (2) Let G be a group and B G be a category with one object whose morphisms B G (, ) = G. Given a group homomorphism f : G H, use it to construct a functor B G B H. (3) Show that the data of a functor B G Sets is the same as that of a G set X. (4) Let C be a groupoid. Prove that C(x, x) is a group under composition. Suppose that x and y are such that C(x, y). Prove that C(x, x) = C(y, y) as groups. (5) Give an example of a category with three objects and exactly five non-identity morphisms. Give an example of a category with two objects and countably many morphisms. Class 3 (1) Let I be the category: 0 1 and F, G : C D. functor: Show that the data of a natural transformation is equivalent to a η : C I D that satisfies η(x, 0) = F (X) and η(x, 1) = G(X). For this reason, a natural transformation is a kind of categorical homotopy. (2) (MacLane) C is a subcategory of C if obj(c) obj(c ) and for any X and Y in C, C(X, Y ) C (X, Y ). It is a full-subcategory if C(X, Y ) = C (X, Y ). Let C be a full subcategory of C, F, G : D C be functors and ι : C C be the functor defined by the inclusion. Prove that Nat(F, G) = Nat(iF, ig) where Nat(F, G) is the set of natural transformations from F to G. (3) What is the pushout in groups, abelian groups and commutative rings? 1

2 2 SOME EXERCISES Class 4 (1) (a) Let C = Ab. Describe the limit and the colimit of a diagram with shape: (These are called equalizers and coequalizers respectively.) (b) In the category of Ab, prove that lim A n n is isomorphic to the kernel of s A n A n n n where s(..., a n,..., a 2, a 1, a 0 ) = (..., a n i(a n+1 ),..., a 2 i(a 3 ), a 1 i(a 2 ), a 0 i(a 1 )). Here, i : A n+1 A n denote the maps in the inverse limit diagram. Describe this an equalizer. (c) In the category of Ab, prove that lim A n n is isomorphic to the cokernel of s A n n n where s(a n ) = a n i(a n ). Here, i : A n A n+1 denote the maps in the direct limit diagram. Describe this as a coequalizer. (2) Describe the colimit of of the following direct systems in the category of abelian groups. (a) Z p Z p Z... Z p Z... A n (b) Z/p p Z/p 2 p Z/p 3... Z/p n p Z/p n+1... (3) Let CX = (X I)/(X {0}). (a) Describe CX as a pushout. (b) Let i : X CX be the map i 1 : X X I, i 1 (x) = (x, 1) followed by the quotient X I CX. Prove that CX. (c) Describe the following pushouts:

3 SOME EXERCISES 3 (i) S 1 i (ii) S 1 i CS 1 (d) Conclude that the pushout construction is not homotopy invariant. (4) Let I = [0, 1] have base point 0. Let P X = X I = Map (I, X) with base point the constant map at the base point of X. Let p 1 : X I X be p 1 (f) = f(1). (a) Describe ΩX = Map (S 1, X) as a pull-back. (b) Prove that P X. (c) Use this to explain why the pull-back construction is not homotopy invariant. (5) (a) Describe X Y as a pushout. (b) Give a bijection Map (X Y, Z) Map (X, Map (Y, Z)). Class 5 (1) Let S n denote the one-point compactification of R n. Explicitly, this is the set R n { } topologized so that the complements of compact sets in R n are open. (You can think of these complements as neighborhoods of ). (a) This is not an abuse of notation, i.e. S n is homeomorphic to the subspace of R n+1 consisting of unit vectors. You can try to show this using stereographic projection if you want, or just skip to the next part. (b) Write down a map S n S m S n+m and show it s a homeomorphism. (This should be easier to do with the one-point compactification definition than with the unit vector definition.) (c) More generally, if V and W are vector spaces, we can define the one-point compactifications S V and S W in the same way. Show that, in this language, we can rewrite (b) as a natural homeomorphism S V S W = S V W. (2) (a) Prove the following lemma:

4 4 SOME EXERCISES Lemma 0.1 (Eckman-Hilton argument). Let X be a set and let, : X X X be two unital binary operations with the same unit e X. Suppose that (a b) (c d) = (a c) (b d) Then = and the operation is both commutative and associative. (b) Let Q be a H cogroup and W be an H group. Prove that the two group multiplications on [Q, W ] satisfy the hypotheses of the Eckman-Hilton argument. Class 6 (1) Prove that if W is an H space, then π 1 W is an abelian group. (2) Check that the map Map (ΣX, Y ) Map (X, ΩY ) defined as f f(x)(t) = f(x t) induces a group isomrophism [ΣX, Y ] [X, ΩY ] which is natural in both X and Y. (3) Let Z be path connected. Suppose that π n 1 Z = 0. For g : Y Z and S n 1 Y, g extends to a map Y f D n. (4) In the following problem, let S n 1 D n be the inclusion of the boundary, X X X X be the map (id ) ( id) and X X X be the fold map id id. For a based topological space X, let J 2 (X) = (X X)/((x, ) (, x)). That is, J 2 (X) is the push-out X X X. X X J 2 (X) Further, you may use the fact that D 2n = I 2n = D n D n and other such standard homeomorphisms without proof. (a) Describe S n S n as a CW-complex obtained from S n S n by attaching a single 2n cell. Exhibit this as a pushout.

5 SOME EXERCISES 5 (b) Use your construction in (b) to give J 2 (S n ) the structure of a CW -complex with one n cell and one 2n cell. (c) Show that S n is an H-space if and only if the attaching map of the 2n-cell of J 2 (S n ) is null-homotopic. Class 8 (1) Let X = {0, 1} with the trivial topology. Let A = {0}. Show that the inclusion A X is a cofibration whose image is not closed. (2) (a) Assume that L is a locally compact topological space. Consider a pushout U g V f k W j W U V. Prove that U L g id V L f id k id W L j id (W U V ) L. is a pushout. (b) Use (a) to conclude that (Y f X) I = (Y I) f id (X I). Class 9 (1) Let X f Y. Prove that Y M f is a cofibration. (2) Let A i X be a cofibration. Prove that A Y i id Y X Y is a cofibration. (3) Let SX = (X I)/(X {0} X {1}) and ΣX = (X I)/(X {0} X {1} I) Prove that if X is well-pointed, the natural map SX ΣX is a homotopy equivalence. Class 11 (1) There are homeomorphism ΣC f = CΣf = C Σf.

6 6 SOME EXERCISES Class 12 p 3 p 2 p 1 f (1) For P p2 Pp1 Pf X Y, prove that the following commutes up to homotopy ΩX Ωf ΩY. P p2 p3 P p1 (2) Check that if E Y X Z is a pull-back, then so is E I Y I X I Z I Class 13 (1) Suppose that E p B is a fibration of unbased spaces. Let P p,b = {(e, α) α(1) = p(e), α(0) = b} E B I be the homotopy fiber over b B. Check that if b 1 and b 2 are in the same path component of B, then P p,b1 is homotopy equivalent to P p,b2. Conclude that the fiber over each points of the same path component are all homotopy equivalent. (2) ( Aguilar et al.) If p : E B is a fibration and B is contractible, prove that there is a homotopy equivalence φ : E B F such that the following diagram commutes E φ B F p B π B Class 15 (1) Define coequalizers as pushouts and colimits of diagrams... as coequalizers. Use this to define homotopy coequalizers and homotopy colimits of diagrams....

7 SOME EXERCISES 7 (2) Prove that if f is a fibration, then the natural map X Z Y E f,g which sends (x, z, y) to (x, α z, y), for α z the constant path at z, is a homotopy equivalence. (3) Define equalizers as pull-backs and limits of diagrams... as equalizers. Use this to define homotopy equalizers and homotopy limits of diagrams.... Class 16 (1) Let A X. An element [f] π q (X, A) is trivial (i.e. homotopic to the constant map at ) if and only if it has a representative f : (D q, S q 1 ) (X, A) such that f(d q ) A. (2) Prove that f : X Y is an n equivalence if and only if (M f, i 1 (X)) is a n connected. Class 18 (1) If A B X and (B, A) and (X, B) have HELP, prove that (X, A) has HELP. (1) Prove that given a commative diagram Class 19 S n 1 D n g f g Y Z e there exists g such that the upper triangle commutes and the lower triangle commutes up to a homotopy relative to S n 1, then π n Y Z is surjective and π n 1 Y Z is injective. Conclude that if this holds for all n, e is a weak equivalence. Class 20 (1) Construct spaces Y n and maps X i Y n such that i induces isomorphisms π π q Y n q X 0 q n = 0 q n + 1. (2) Any two maps f : X Y between CW complexes is homotopic to a cellular map. Any homotopic cellular maps are homotopic via a cellular homotopy, that is, a homotopy H : X I Y which is a cellular map. Class 22 (1) Prove that if X is n connected, then ΣX is n + 1 connected.

8 8 SOME EXERCISES (2) (See Concise, Chapter 10, Section 7) Identify maps that will make the following diagram commutative up to homotopy: ΣΩP f ΣΩX ΣΩY ΣP f ΣX ΩY P f X Y C f ΣX ΩY ΩC f ΩΣX ΩΣY ΩΣC f