Chapter 12 — Three Computational Axes

Open source document →

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:

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

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

   $$0\to C^k(K_{\alpha },L_{\alpha };A)\stackrel{J_k}{\to }{C}^k(K_{\alpha };A)\stackrel{R_k}{\to }{C}^k(L_{\alpha };A)\to 0$$
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:

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

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

   $$\mathrm{Barcode}(\Sigma ):=\{[a_i,b_i)\subset {\mathbb{R}}_{\ge 0}\}$$
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

   $${\dim }_H(\Sigma ):=\inf \{\delta >0:\sum _i{\ell }_i^{\delta }<\infty \}$$
10. #10

    If is finite and :

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

    For each degree , the cochain short exact sequence

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

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

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

    Exact sequence check ( ):

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