--- date: 2022-12-25 04:07 modification date: Sunday 25th December 2022 04:07:12 title: "Algebraic Topology" flashcard: "Quals::Topology" --- # Point-Set ## Definitions - [ ] What is a retract? - [ ] What is a locally compact space? - [ ] What is a normal space? - [ ] What is a Hausdorff space? - [ ] What is a semilocally simply connected space? - [ ] What is a local homeomorphism? - [ ] What is a first countable space? - [ ] What is a neighborhood basis? - [ ] What is a limit point? - [ ] What is a deformation retract? - [ ] What is a sequentially compact space? ## Exercises - [ ] When does sequential compactness coincide with compactness? - [ ] Are singletons open or closed? - [ ] Show that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. - [ ] Give an example of a function $f: \RR^n \to \RR$ that is continuous in each variable but not continuous. ## Results - [x] What is the Euler characteristic in terms of genus? ✅ 2023-01-24 - [ ] $\chi(\Sigma_g) = 2-2g$ if $\Sigma_g$ is orientable, $\chi(M_g) = 2-g$ if $M_g$ is nonorientable. - [ ] Puncturing reduces $\chi$ by 1. - [x] $\chi(\Sigma) = 2 \implies\cdots$ ✅ 2023-01-24 - [ ] $g=0$ so $\Sigma \cong S^2$. - [x] $\chi(\Sigma) = 0 \implies\cdots$ ✅ 2023-01-24 - [ ] $g=1$ so $\Sigma \cong T^1$ (no boundary, orientable), or - [ ] $K$ the Klein bottle (no boundary, nonorientable). - [ ] $M$ the Mobius band (1 boundary, nonorientable), or - [ ] $S^2\smts{\pt_1, \pt_2} \cong S^1\times I$ an annulus (2 boundaries), or - [x] $\chi(\Sigma) = -2 \implies\cdots$ ✅ 2023-01-24 - [ ] $g=2$ so $\Sigma\cong \Sigma_2$ is a 2-holed torus, or - [ ] $S^2\smts{\pt_1,\cdots, \pt_4}$. - [ ] What is Urysohn's lemma? - [ ] What is the tube lemma? - [x] What is $\chi(A\# B)$? ✅ 2023-01-24 - [ ] $\chi(A) + \chi(B) - 2$ # Algebraic Topology ## Sequences - [x] Long Exact Sequence of a Pair $(A, B)$ ✅ 2023-01-24 - [ ] $$\ldots H_n(B) \to H_n(A) \to H_n(A,B) \to H_{n-1}(B) \ldots$$ - [x] UCT: Change of Coefficients SES ✅ 2023-01-24 - [ ] $${\displaystyle \rm{Tor}_\ZZ^0 (H_{i}(X;\ZZ), A)\,{\injects }\,H_{i}(X;A)\surjects \rm{Tor}_\ZZ^1 (H_{i-1}(X;\ZZ ),A)}$$ - [x] UCT: Cohomology and Homology SES ✅ 2023-01-24 - [ ] $${ \rm{Ext}_{\ZZ}^{1}(H_{i-1}(X; \ZZ),R)\injects H^{i}(X; R)\surjects \rm{Ext}_{\ZZ}^{0}(H_{i}(X; \ZZ),R) }$$ - [x] Kunneth SES ✅ 2023-01-24 - [ ] $$ \bigoplus_{i+j=k}H_{i}(X;R)\otimes _{R}H_{j}(Y;R)\injects H_{k}(X\times Y;R)\surjects \bigoplus_{i+j=k-1}{\rm {Tor}}_{R}^{1}(H_{i}(X;R),H_{j}(Y;R))$$ - [x] Kunneth isomorphism (nice case) ✅ 2023-01-24 - [ ] $$H_{k}(X\times Y;R) \cong \bigoplus_{i+j=k}H_{i}(X;R)\otimes H_{j}(Y;R)$$ - [x] Cohomology in terms of homology (nice case) ✅ 2023-01-24 - [ ] $$H^i(X; \ZZ) = \Free(H_i(X; \ZZ)) \oplus \tors(H_{i-1}(X; \ZZ))$$ - [x] Homology in terms of cohomology (nice case) ✅ 2023-01-24 - [ ] $$H_i(X; \ZZ) = \Free(H^i(X; \ZZ)) \oplus \tors(H^{i+1}(X; \ZZ))$$ - [ ] Mayer-Vietoris sequence? ## Computations - [x] Prove $\Hom_R(R, A) \cong A$. ✅ 2023-01-24 - [ ] Take the map $\Phi$ where $f\mapsto f(1)$, - [ ] Surjects by defining $f_a(1) \da a$ for any $a\in A$ and extending by $f_a(n) \da na$ - [ ] Injects: if $f(1) = g(1) = a$ then $f(n) \da na$ and $g(n) \da na$ so $f\equiv g$. - [ ] The tor complex between $C_n$ and $A$. - [ ] $$\Tor^\ZZ_*(C_n, A) = (A/nA)\cdot t^0 \oplus A[n]\cdot t^1 \oplus \cdots$$ - [ ] The ext complex between $C_n$ and $A$. - [ ] $$\Ext^*_\ZZ(C_n, A) = A[n]\cdot t^0 \oplus (A/nA) \cdot t^1 \oplus \cdots$$ - [ ] Hom Table - [ ] First row: $\Hom_\ZZ(C_n, A)\cong A[n]$ - [ ] Second row: $\Hom_\ZZ(\ZZ, \wait) \cong \id$ - [ ] $$\begin{array}{c|c|c|c} \Hom_\ZZ & C_m & \ZZ & \QQ \\\hline C_n & C_m[n] = C_{\gcd(m, n)} & \ZZ[n] = 0 & \QQ[n] = 0 \\\hline \ZZ & \id(C_m) = C_m & \id(\ZZ) = \ZZ & \id(\QQ) = \QQ \\\hline \QQ & 0 & 0 & \QQ\end{array}$$ - [ ] Ext Table - [ ] Use $\Ext^1_\ZZ(C_n, A) = A/nA$. - [ ] Use $\Ext(\wait, \QQ) = 0$ since $\QQ$ is injective and $\Ext(\ZZ, \wait) = 0$ since $\Hom_\ZZ(\ZZ, \wait) = \id$. - [ ] $$\begin{array}{c|c|c|c} \Ext^1_\ZZ & C_m & \ZZ & \QQ \\\hline C_n & C_m/nC_m = C_{\gcd(n, m)} & \ZZ/n\ZZ = C_n & \QQ/n\QQ = 0 \\\hline \ZZ & 0 & 0 & 0 \\\hline \QQ & 0 & \mca^\fin/\QQ & 0 \end{array}$$ | $\Ext^*_\ZZ(V, H)$ | $C_m$ | $\ZZ$ | $\QQ$ | |:-------------------|:----------------------------------|:------------------------------------|:--------------------------------| | $C_n$ | $C_m[n] + C_m/nC_mt = C_d + C_dt$ | $\ZZ[n] + \ZZ/n\ZZ t = C_n + C_n t$ | $\QQ[n] + \QQ/n\QQ[t] = 0 + 0t$ | | $\ZZ$ | $C_m + 0 t$ | $\ZZ + 0t$ | $\QQ + 0t$ | | $\QQ$ | $0+0t$ | $0 + (\mca^\fin/\QQ)t$ | $\QQ + 0t$ | - [ ] Full tor complex table: - [ ] E.g. top-left is $\Tor_*^\ZZ(C_n, C_m)$. - [ ] First row: $C_n\tensor_\ZZ A = A/nA$ and $\Tor_1^\ZZ(C_n, A) = A[n]$. - [ ] Second row: $\ZZ\tensor_\ZZ A = A$ and $\Tor_1^\ZZ(\ZZ, A) = 0$ since $\ZZ$ is projective. - [ ] Third row: $G\tensor_\ZZ \QQ = \QQ\tensor_\ZZ G = 0$ for $G$ any group - [ ] Also $\Tor(A, B) = \Tor(B, A)$. | $\Tor_*^\ZZ(V, H)$ | $C_m$ | $\ZZ$ | $\QQ$ | |:---------------------|:-----------------------------------------------|:----------------------------------------------|:------------------------------------------| | $C_n$ | $C_m/nC_m \oplus C_m[n]t = C_d \oplus C_d t$ | $\ZZ/n\ZZ \oplus \ZZ[n]t = C_n \oplus C_nt$ | $\QQ/n\QQ \oplus \QQ[n]t =0\oplus 0t$ | | $\ZZ$ | $\ZZ/m\ZZ + \ZZ[m]t = C_m + C_m t$ | $\ZZ + 0t$ | $\QQ + 0t$ | | $\QQ$ | $\QQ/n\QQ + \QQ[n]t = 0 + 0t$ | $\QQ + 0t$ | $0 + 0t$ | ## Examples - [x] Homology of real projective space, $H_* \RP^2$ ✅ 2023-01-24 - [ ] $$H_* \RP^2 = \ZZ\cdot t^0 + C_2 \cdot t^1$$ - [ ] $H_*(K)$ for $K$ the Klein bottle. - [ ] $H_*(\CP^n)$