1 Chain Complexes Poincar´e1900 Simplicial Homology. Cn = free abelian group on the n-simplex with ordered P vertices, and d : Cn ! Cn−1 sends an n-simplex to τ2@σ ±τ. The incidence matrix has every entry 0; +1; −1 depending on if the vertex is on the simplex and what the orientation is. 2 5 ..................................................................................................................................................... .. •... .....• ... ..... ... ..... ... ..... ... ..... ... .... ... ..... ... ..... ... ..... ... ..... ... ..... ... .... ... ..... ... ..... ........ ...... ..... ... ..... ... .... ... ..... ... ..... ... 1..... ... 4 ..... ... .................................................................................................................................................... ... .. • ... .....• ... ..... ... ..... ... .... ... ..... ... ..... ........ ...... ..... ... ..... ... .... ... ..... ... ..... ... ..... ... ..... ... ..... ... ..... ... ..... ... ..... ... ..... ... ..... ... 0..... ...3 •....................................................................................................................................................... .•. 2 0 0 0 −1 −1 0 3 6 −1 0 0 0 0 −1 7 6 7 6 0 −1 −1 0 0 0 7 This graph has incidence matrix 6 7 where 6 0 0 1 1 0 0 7 6 7 4 1 0 0 0 1 0 5 0 1 0 0 0 1 the rows are indexed by the edges and the columns are indexed by the edges, with a 1 if an edge ends at that point, −1 is it begins there. Fir a connected graph with V vertices and E edges, d has rank V − 1 and E V has a splitting, so Z = C1 ! C0 = Z with H0(graph) ' Z and H1(graph) ' ZE−V −1. Definition 1.1 (Chain Complex). A chain complex is a set of objects fCng in a category like vector spaces, abelian groups, R-mod, and graded R-mod, with dn : Cn ! Cn−1 maps such that the kernel of dn is Zn, the n-cycles of C∗, the image of dn+1 is Bn, the n-boundaries of C∗ and Hn(C) = Zn=Bn is the kernel of the dn(coker dn+1) An Ab-category is a category such that the Hom-sets are abelian groups and composition is bilinear. An additive category is an Ab-category where finite products and coproducts exist and are the same. f ι Monics vs. Kernels - ι monic=cancellation, X ! Y ⊆ Z, that is, ιf = 0 ) f = 0 ker(f) is a universal defined by the following diagram: 1 . K ................ ....... .............. ....... .............. ...... ....... .............. ... ....... .......0....... ....... .............. ....... ............. ........ .............. ........ ................ f ..... ............................................................................................................ .. .. 9! . .. ... XY............... ......... .............. ....... .............. ....... .............. 8....... 0............. ....... .............. ....... .............. ....... .............. ................. A ... f Epis vs. Cokernels - π epi=cancellation, X Y π Z. Epis are the dual of monics and cokernels are the duals of kernels. Im(f) = ker(coker f) Definition 1.2 (Abelian Category). An abelian category A is an additive cat- egory in which monics=kernels, epis=cokernels, every monic ι : A ! B is the kernel of the map from B to coker(ι) and every epi π : B ! C is the cokernel of ker(π) ! B. Theorem 1.1. Ch=the category of chain complexes in A is an abelian cate- gory. f Proof. A morphism C∗ ! D∗ of chain complexes is a family of maps fn : Cn ! Dn which commute with d, that is all squares below commute: d .... d .... d .... ::: ............................................................................................................. Cn ................................................................................................... Cn−1 ................................................................................................... ::: . f . f . ....... ....... ..... ..... d .... d .... d .... ::: ............................................................................................................. Dn ................................................................................................... Dn−1 ................................................................................................... ::: It is an Ab-category because it inherits the addition of morphisms, products and coproducts are done termwise, and we set the nth term of the kernel to the kernel of the nth term morphism, similarly with cokernels and images. Operations p Shift - C[p] is the complex which in degree n is Cn+p, dC[p] = (−1) dC 8 < 0 n < p Good Truncation - (τ≥pC) is the complex which in degree n is Zp n = p : Cn n > p ... ... ... ... ::: ....... ...................................................................................................... 0 ....... ...................................................................................................... Zp ....... .............................................................................................. Cp+1 ....... .............................................................................................. ::: . .... ... ... ....... ....... ....... ..... ... ..... ... ... ... ... ::: ....... ............................................................................................ Cp−1 ....... ............................................................................................ Cp ....... .............................................................................................. Cp+1 ....... .............................................................................................. ::: 0 n < p Then Hn(τC) = Hn(C) n ≥ p 0 n < p Brutal Truncation - (β≥pC)n = and Hp(βC) = Cp=Bp is Cn n ≥ p different. 2 H C n < p And the dual notions, τ C = C/τ C has H (τ C) = n <p ≥p n <p 0 n ≥ p and β<pC = ker(C ! β≥pC) has Hp−1(β<oC = Zp−1 is ::: Cp−2 Cp−1 0 is like a p-skeleton. Definition 1.3 (Double Complex C∗∗). A double complex C∗∗ is a Z×Z indexed h b h h v v family of objects Cpq and maps d ; d such that d ◦ d = 0, d ◦ d = 0 and dh ◦ dv + dv ◦ dh = 0. v . d . ....... ..... ... ... Cp+1;q ....... ............................................................................... Cp;q ....... ............................................................................... Cp−1;q h h . d . d . v . h . d . d . ....... ....... ..... ..... ... Cp+1;q−1 ....... ...................................................... Cp;q−1 dh Operations ⊕ L v h Total Complex - (Tot )n = p+q=n Cp;q, d = d + d . Definition 1.4 (Bounded Double Sequence). A double sequence is said to be bounded if 8n only finitely many Cpq with p + q = n are nonzero. Q Q Variant: (Tot )n = p+q=n Cp;q is the product total complex. The product total complex and the total complex are the same iff C is bounded. Example: h v Cpq = Z=4 for every p; q and d = d = 2. 3 . ....... ....... ....... ..... ..... ..... ... ... ... ... ::: ....... ................................................................................................... Z=4 ....... ................................................................................................. Z=4 ....... ................................................................................................. Z=4 ....... ................................................................................................... ::: . ....... ....... ....... ..... ..... ..... ... ... ... ... ::: ....... ................................................................................................... Z=4 ....... ................................................................................................. Z=4 ....... ................................................................................................. Z=4 ....... ................................................................................................... ::: . ....... ....... ....... ..... ..... ..... ... ... ... ... ::: ....... ................................................................................................... Z=4 ....... ................................................................................................. Z=4 ....... ................................................................................................. Z=4 ....... ................................................................................................... ::: . ....... ....... ....... ..... ... ... Definition 1.5 (Acyclic). A complex C∗ is acyclic if Hn(C) = 0 for all n. Definition 1.6 (Exact). A sequence Cn−1 Cn Cn+1 is exact if C∗ is acyclic. Compare to a chain complex of chain complexes. We have A an abelian category, and so we know that Ch(A) is an abelian category, and so we can take Ch(Ch(A)) th p object Dp = Cp;∗ and fp : Dp ! Dp−1 is a morphism of complexes, so it is really fpq : Cpq ! Cp−1;q. 4 . ....... ....... ..... ..... Cp−1;q−1 Cp;q+1 . ....... ....... ..... ..... ... ... Cp−1;q ....... ............................................................................... Cp;q ....... ............................................................................... Cp+1;q . f . f . ....... ....... ..... ..... ... Cp−1:q−1 ....... ...................................................... Cp;q−1 f BUT: fd = df so it is not quite a double complex. We use \the sign trick" v p d = (−1) dC to make it into a double complex. Proposition 1.2. The category of double complexes is is equivalent to the cat- egory Ch(Ch(A)) via the sign trick. Aside: There is a \geometric realization" functor from Ch≥0(Ab) to spaces, C 7! jCj Short Exact Sequences of complexes $ fibrating. Acyclic complex $ contractible space ΩjC[−1]j