We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy equivalence, tubular neighbourhood, and spinning can be encoded on a computer and used to calculate ambient isotopy invariants of continuous embeddings $N\hookrightarrow M$ of one topological manifold into another. More specifically, we describe an algorithm for computing the homology $H_n(X,A)$ and cohomology $H^n(X,A)$ of a finite connected CW-complex X with local coefficients in a $\mathbb{Z}{\pi }_{1}X$-module A when A is finitely generated over $\mathbb{Z}$. It can be used, in particular, to compute the integral cohomology $H^n({\tilde{X}}_H,\mathbb{Z})$ and induced homomorphism $H^n(X,\mathbb{Z})\to H^n({\tilde{X}}_H,\mathbb{Z})$ for the covering map $p:{\tilde{X}}_H\to X$ associated to a finite index subgroup $H<{\pi }_{1}X$, as well as the corresponding homology homomorphism. We illustrate an open-source implementation of the algorithm by using it to show that: (i) the degree 2 homology group $H_2({\tilde{X}}_H,\mathbb{Z})$ distinguishes between the homotopy types of the complements $X\subset {\mathbb{R}}^4$ of the spun Hopf link and Satoh's tube map of the welded Hopf link (these two complements having isomorphic fundamental groups and integral homology); (ii) the degree 1 homology homomorphism $H_1(p^{-1}(B),\mathbb{Z})\to H_1({\tilde{X}}_H,\mathbb{Z})$ distinguishes between the homeomorphism types of the complements $X\subset {\mathbb{R}}^3$ of the granny knot and the reef knot, where $B\subset X$ is the knot boundary (these two complements again having isomorphic fundamental groups and integral homology). Our open source implementation allows the user to experiment with further examples of knots, knotted surfaces, and other embeddings of spaces. We conclude the paper with an explanation of how the cohomology algorithm also provides an approach to computing the set ${[W,X]}_{\varphi }$ of based homotopy classes of maps $f:W\to X$ of finite CW-complexes over a fixed group homomorphism ${\pi }_{1}f=\varphi$ in the case where $\dim W=n$, ${\pi }_{1}X$ is finite and ${\pi }_{i}X=0$ for $2\le i\le n-1$.

我们展示了如何在计算机上编码具有局部系数，CW复杂度，覆盖空间，同胚性等价，简单同伦等价，管状邻域和旋转的经典同调概念，并可以用于计算连续嵌入的环境同位素不变量 $ñ\hookrightarrow \mathrm{中号}$一个拓扑流形到另一个拓扑流形。更具体地说，我们描述了一种用于计算同源性的算法$H_{ñ}（X，\mathrm{一种}）$ 和同调 $H^{ñ}（X，\mathrm{一种}）$ 有限连接的CW复数X在a中具有局部系数 $\mathbb{ž}{\pi }_{\mathrm{1个}}X$在A上有限生成A时的-module A$\mathbb{ž}$。它尤其可以用于计算积分同调$H^{ñ}（{\stackrel{〜}{X}}_H，\mathbb{ž}）$ 并诱发同态 $H^{ñ}（X，\mathbb{ž}）\to H^{ñ}（{\stackrel{〜}{X}}_H，\mathbb{ž}）$ 用于覆盖图 $p：{\stackrel{〜}{X}}_H\to X$ 与有限索引子组关联 $H<{\pi }_{\mathrm{1个}}X$，以及相应的同源性。我们通过使用以下算法来说明该算法的开源实现：（i）2级同源性组$H_{\mathrm{2个}}（{\stackrel{〜}{X}}_H，\mathbb{ž}）$ 区分补体的同型异型 $X\subset {\mathbb{[R}}^4$旋转的Hopf链环和焊接的Hopf链环的Satoh管图（这两个补体具有同构的基团和整体同源性）；（ii）1级同源性$H_{\mathrm{1个}}（p^{-\mathrm{1个}}（乙），\mathbb{ž}）\to H_{\mathrm{1个}}（{\stackrel{〜}{X}}_H，\mathbb{ž}）$ 区分补体的同胚型类型 $X\subset {\mathbb{[R}}^3$ 奶奶结和礁结的地方 $乙\subset X$是结边界（这两个补语又具有同构的基团和整数同源性）。我们的开放源代码实现允许用户尝试打结，打结的表面以及其他空间嵌入的其他示例。我们在本文结尾处解释了同调算法如何也提供了一种计算集合的方法${[\mathrm{w\; ^}，X]}_{\varphi }$ 地图的同伦分类 $F：\mathrm{w\; ^}\to X$ 固定群同态上有限CW络合物的分布 ${\pi }_{\mathrm{1个}}F=\varphi$ 在这种情况下 $\mathrm{暗淡}\mathrm{w\; ^}=ñ$， ${\pi }_{\mathrm{1个}}X$ 是有限的 ${\pi }_{\mathrm{一世}}X=0$ 为了 $\mathrm{2个}\le \mathrm{一世}\le ñ-\mathrm{1个}$。