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Exercises for Math 276

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Exercises for Math 276 Exercises for Math 276 Robert Kropholler April 30, 2020 These exercises will be updated periodically throughout the course of the semester. You do not have to attempt all the exercises. They are here to help cement the concepts discussed in lectures and to clarify some points made in lectures. Please inform me if you find any errors or have any questions about the exercises. 1 Exercises due Feb 21st 1. Let S be a set. The free Abelian group on S is a pair pF pSq; iSq consisting of an Abelian group F pSq and an function iS : S Ñ F pSq such that for any Abelian group A and map j : S Ñ A there is a unique homomorphism φ: F pSq Ñ A such that φ ˝ iS “ j. (a) Find a map iS : S Ñ `SZ, such that the above property holds. 1 1 (b) Suppose that pF pSq; iSq and pF pSq ; iSq are free abelian groups on S. Show that there 1 1 is an isomorphism φ: F pSq Ñ F pSq such that φ ˝ iS “ iS. (c) Show that we can extend the function S ÞÑ F pSq to a functor SetÑAb. (d) Show that X ÞÑ CnpXq can be extended to a functor TopÑAb. 2. Let σ : I Ñ X. Show thatσ ¯ „ ´σ. Whereσ ¯ptq “ σp1 ´ tq: 3. Let A be a subspace of X and i: A Ñ X the inclusion map. Show that i extends to an injective homomorpshism CnpAq Ñ CnpXq. 4. Suppose r : X Ñ A is a retraction. I.e. r ˝ i: A Ñ A is the identity. (a) Show that r˚ : HnpXq Ñ HnpAq is surjective. (b) Show that i˚ : HnpAq Ñ HnpXq is injective. (c) Find spaces A; X such that A Ă X and for some n i˚ : HnpAq Ñ HnpXq is not injective. (Hint: look at the case n “ 0.) 5. (optional) Let S be obtained by taking a disjoint union of 2-simplices and identifying edges in pairs, where every edge is in a unique pair. Show that S is locally homeomorphic to R2. 6. (a) Show that if two loops are homotopic relative to t0; 1u, then they are homologous. (b) Deduce that there is a homomorphism π1pX; bq Ñ H1pXq. (c) Following the ideas from class show that if X is path connected, then the map above is surjective. 1 7. (optional) Let V be a totally ordered finite set. Let L “ pV; Σq be a simplicial complex. Let SnpLq be the free abelian group on the set trv0; : : : ; vns | vi P V; tv0; : : : ; vnu P Σ; vi ă vj if i ă n i ju. Let δn : SnpLq Ñ Sn´1pLq be given by δnprv0; : : : ; vnsq “ i“0p´1q rv0;:::; v^i; : : : ; vns: (a) Show that δn ˝ δn`1 “ 0. ř ∆ (b) Define the n-th simplicial homology group Hn pLq “ kerpδnq{Impδn`1q. Show that if L ∆ is an n-dimensional simplicial complex then Hk pLq “ 0 for all k ¡ n. (c) Show that if L is an n-dimensional simplicial show that HnpLq is free-abelian. (d) Pick a triangulation for S1 and S2, compute their simplicial homology. It is worth noting that simplicial homology is isomorphic to singular homology. A proof can be found in Hatcher, we will not study it in this courses for the issues mentioned in lectures. 2 Exercises due 28th Feb 1. Show that any long exact sequence can be broken up into short exact sequences. 2. Let DnpX; Aq be the free abelian group on the singular n-simplices σ such that Impσq Ć A. (a) Show that CnpXq “ CnpAq ` DnpX; Aq. (b) Show that CnpAq X DnpX; Aq “ 0. (c) Show that DnpX; Aq – CnpX; Aq. The isomorphism is the restriction of the projection CnpXq Ñ CnpX; Aq. (d) Deduce that the short exact sequence 0 Ñ CnpAq Ñ CnpXq Ñ CnpX; Aq Ñ 0 splits. (e) Why isn't this splitting a chain map? 3. Finish the proof of the snake lemma by showing that i˚δ˚ “ 0 and the sequence is exact at both HnpA˚q and HnpC˚q. 4. Show that if C is free, then the short exact sequence 0 Ñ A Ñ B Ñ C Ñ 0 splits. 5. Show that the short exact sequence 0 Ñ Z Ñ Z Ñ Z{2Z Ñ 0 does not split. 6. (optional) Show that if a short exact sequences 0 Ñ A Ñ B Ñ C Ñ 0 splits. Then B – C `A. n 7. (optional) Let G be an Abelian group. Define CnpX; Gq “ t σ gσσ | σ : ∆ Ñ X; gσ P Gu. This is a group with pointwise addition and is isomorphic to `σG. Define B : CnpX; Gq Ñ ř Cn´1pX; Gq as for singular homology. Note ´g still makes sense since G is an Abelian group. We define the homology of X with coefficients in G to be HnpX; Gq “ HnpC˚pX; Gqq. (a) Show that Hnppt; Gq “ G if n “ 0 and is trivial otherwise. (b) Show that if 0 Ñ G1 Ñ G Ñ G2 Ñ 0 is a short exact sequence of Abelian groups. Then 1 2 the induced sequence 0 Ñ CnpX; G q Ñ CnpX; Gq Ñ CnpX; G q Ñ 0 is also exact. (c) Deduce that there is a long exact sequence in homology with coefficients. In the special case that G1 “ G “ Z and G2 “ Z{pZ the connecting homomorphism is known as the Bockstein homomorphism. 8. (optional) 2 (a) Show that the fundamental group of the Klein bottle G “ xx; y | x´1yx “ y´1y fits into a short exact sequence 0 Ñ Z Ñ G Ñ Z Ñ 0. (b) Show that the above sequence splits. (c) Show that G is not Abelian. (d) Deduce that for non-abelian splitting does not mean G “ Z ˆ Z. (e) Look up the definition of a semi-direct product to see what happens in the non-Abelian case. 3 Exercises due March 6th 1. (a) Show that there is no retraction from Dn Ñ Sn´1. (b) Show that every map from Dn Ñ Dn has a fixed point. 2. (a) Show that Sn is not homotopy equivalent to Sm if n ‰ m. (b) Show that Rn is not homeomorphic to Rn if n ‰ m. 3. Finish the proof of the 5-lemma by showing that the map f3 is surjective. 4. (optional) Let A Ă B Ă C Ă X be topological spaces. Assume B is open and B¯ Ă intpCq. Assume that there is a deformation retraction r : C Ñ A, i.e. r ˝ i “ idA and there is a homotopy H : C ˆ I Ñ C such that Hpx; 0q “ x; Hpx; 1q “ rpxq and Hpa; tq P A for all a P A; t P I. Note this implies that C is homotopy equivalent to A. (a) Show that HnpX; Aq – HnpX; Cq. (b) Show that there is a deformation retraction C{A Ñ A{A. (You may assume that the map C ˆ I Ñ C{A ˆ I given by px; tq ÞÑ prxs; tq is a quotient map.) (c) Show that there is an isomorphism HnpX{A; C{Aq Ñ HnpX{A; A{Aq “ H~npX{Aq. (d) Show that there is a homeomorphism of pairs pX rB; C rBq Ñ ppX rBq{A; pC rBq{Aq. (e) Excise B and B{A to complete the proof that HnpX; Aq – H~npX{Aq. 5. Show that HnpX; X r txuq only depends on a closed neighborhood U of x. 6. Let ConepXq be the topological space X ˆ I{px; 0q „ px1; 0q for all x; x1 P X. Show that ConepXq is contractible for any X. 7. Let X be a non-empty topological space. Define the suspension ΣpXq to be Cone´pXqYXˆt1u Cone`pXq, where Cone´pXq “ Cone`pXq “ ConepXq. (a) Use the excision and homotopy axioms to show that HnpΣpXq; Cone´pXqq – HnpCone`pXq;Xq: ~ (b) Use the long exact sequence to show that HnpΣpXqq – HnpΣX; Cone´pXqq. ~ (c) Use the long exact sequence to show that HnpCone`pXq;Xq – Hn´1pXq. ~ (d) Deduce that HnpΣpXqq – Hn´1pXq. 3 CnpXq; if n ¥ 0, 8. (optional) Recall the reduced chain complex of a space X is C~npXq “ #Z; if n “ ´1. Where the map C0pXq Ñ Z is given by σ nσσ ÞÑ σ nσ. Let A Ă X. Then define ι: C~ pAq Ñ C~ pXq as before for n ¥ 0 and an isomorphism for n “ ´1. n n ř ř (a) Show that ι defines a chain map. (b) Define C~npX; Aq as C~npXq{C~npAq. Define the reduced homology of a pair H~npX; Aq “ HnpC~npX; Aq. Show that H~npX; Aq – HnpX; Aq for all n. (c) Deduce that there is a long exact sequence for reduced homology. 9. (optional) Let X; Y be spaces and A Ă X, B Ă Y . Show that HnpX \Y; A\Bq – HnpX; Aq` HnpY; Bq. 0 10. Let x P X and y P Y . Let S “ tx; yu Ă X \ Y . Show that the homomorphism H1pX \ 0 0 Y; S q Ñ H0pS q is always 0. 4 Exercises due March 13th 1. Let X and Y be CW-complexes, describe how to get a cell structure on X ˆ Y . Hint: The cells in X ˆ Y are products Dn ˆ Dm for cells in Dn in X and Dm in Y . 2. Use the above to check the cell structure on the torus by giving a CW structure to S1.
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