Chapter 12 — Three Computational Axes

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1. #1

   The sequential protocol extends:

   $$\mathcal{W}_t\Rightarrow\mathfrak{F}_t\Rightarrow\Sigma_t\Rightarrow\text{Axis 1}\Rightarrow\text{Axis 2}\Rightarrow\text{Axis 3}.$$
2. #2

   $$H^k(K_\alpha;A)=\ker D_K^k/\mathrm{im}\,D_K^{k-1}$$
3. #3

   For each degree , there is a long exact sequence:

   $$\cdots\to H^k(K_\alpha,L_\alpha;A)\xrightarrow{J_k^*}H^k(K_\alpha;A)\xrightarrow{R_k^*}H^k(L_\alpha;A)\xrightarrow{\delta}H^{k+1}(K_\alpha,L_\alpha;A)\to\cdots$$
4. #4

   For each degree , there is a long exact sequence: Proof. At cochain level, the short exact sequence

   $$0\to C^k(K_\alpha,L_\alpha;A)\xrightarrow{J_k}C^k(K_\alpha;A)\xrightarrow{R_k}C^k(L_\alpha;A)\to 0$$
5. #5

   For the relative coboundary matrix (where injects relative basis into and projects to relative basis):

   $$\dim H^k(K_\alpha,L_\alpha;A)=\dim\ker D_{\mathrm{rel}}^k-\mathrm{rank}(D_{\mathrm{rel}}^{k-1}).$$
6. #6

   Under Hypothesis 11.1 (continuous enrichment), excision and retraction give:

   $$H^*(K_\alpha,L_\alpha;A)\cong H^*(M_\alpha\setminus S_\alpha;A).$$
7. #7

   The barcode of the door set is the persistence diagram of the energy-filtered simplicial complex:

   $$\mathrm{Barcode}(\Sigma):=\{[a_i,b_i)\subset\mathbb{R}_{\ge0}\}$$
8. #8

   $$\dim H^k(L_\alpha;A)=|\{i:\text{bar }[a_i,b_i)\text{ is a }k\text{-degree feature with }b_i=\infty\}|.$$
9. #9

   $$\dim_H(\Sigma):=\inf\Bigl\{\delta>0:\sum_i\ell_i^\delta<\infty\Bigr\}$$
10. #10

    If is finite and :

    $$\check H^0(\Sigma;U(1))=U(1)^n,\qquad\check H^{k>0}(\Sigma;U(1))=0.$$
11. #11

    For each degree , the cochain short exact sequence

    $$0\to C^k(K_\alpha,L_\alpha;A)\xrightarrow{J_k}C^k(K_\alpha;A)\xrightarrow{R_k}C^k(L_\alpha;A)\to 0$$
12. #12

    For each degree , the cochain short exact sequence induces the long exact sequence in cohomology:

    $$\cdots\to\underbrace{H^k(K_\alpha,L_\alpha)}_{\text{Axis 2}}\to\underbrace{H^k(K_\alpha)}_{\text{Axis 1}}\to\underbrace{H^k(L_\alpha)}_{\text{Axis 3}}\xrightarrow{\delta}\underbrace{H^{k+1}(K_\alpha,L_\alpha)}_{\text{Axis 2}}\to\cdots$$
13. #13

    Exact sequence check ( ):

    $$0\to H^0(K,L)\to H^0(K)\to H^0(L)\xrightarrow{\delta}H^1(K,L)\to\cdots$$