# Related constructions and facts

There are many homological tools used in algebraic topology. Some variants of the homology groups include:

• Homology group with coefficients: for an abelian group $${G}$$, $${H_{n}\left(X;G\right)}$$ is defined using $${n}$$-chains $${C_{n}\left(X;G\right)}$$ with coefficients in $${G}$$ instead of $${\mathbb{Z}}$$
• Reduced homology groups: a slight variant that avoids the result $${H_{0}=\mathbb{Z}}$$ for points while keeping the higher homology groups the same
• Local homology of $${X}$$ at $${A}$$: $${H_{n}\left(X\mid A\right)\equiv H_{n}\left(X,X-A\right)}$$ depends only on a neighborhood of $${A}$$ in $${X}$$
• Simplicial homology, cellular homology, etc.: more easily constructed homology theories that are only valid for certain types of spaces; all can be shown to be equivalent to singular homology for those spaces
• Cohomology groups: $${H^{n}\left(X;G\right)}$$ are dual constructions based on the cochain groups $${C^{n}\left(X;G\right)\equiv C_{n}^{*}=\textrm{Hom}\left(C_{n},G\right)}$$, the group of homomorphisms from $${C_{n}}$$ to some abelian group $${G}$$; a homomorphism $${H^{n}\left(X;G\right)\to\mathrm{Hom}\left(H_{n}\left(X;G\right),G\right)}$$ can be constructed which is surjective, becoming an isomorphism if $${G}$$ is a field
• Cohomology ring: $${H^{*}\left(X;R\right)}$$ is a direct sum of the cohomology groups $${H^{n}\left(X;R\right)}$$ with coefficients in a ring $${R}$$; multiplication is defined using the cup product, a product between the $${H^{n}\left(X;R\right)}$$

Some related constructions include:

• Betti number: $${b^{n}\equiv}$$ the number of $${\mathbb{Z}}$$ summands if $${H_{n}(X)}$$ is written $${\mathbb{Z}\oplus\dotsb\oplus\mathbb{Z}_{c_{1}}\oplus\mathbb{Z}_{c_{2}}\oplus\mathbb{Z}_{c_{3}}\oplus\dotsb}$$, where the $${c_{i}}$$ are called torsion coefficients; in a cell complex, $${b^{n}}$$ is just the number of $${n}$$-cells
• Euler characteristic: the alternating sum of Betti numbers $${\chi=b^{0}-b^{1}+b^{2}-\dotsb}$$, i.e. for a cell complex the number of even cells minus the number of odd cells; thus a compact connected surface has genus $${g=\left(2-\chi\right)/2}$$
• Brouwer degree (AKA winding number for $${S^{1}}$$): any mapping $${\phi\colon S^{n}\to S^{n}}$$ induces a homomorphism on $${H_{n}(S^{n})=\mathbb{Z}}$$ of the form $${z\rightarrow az}$$; the integer $${a}$$ is the Brouwer degree of the map, essentially the number of times the mapping wraps around the sphere
• Moore space: given an abelian group $${G}$$ and an integer $${n>0}$$, the space $${M\left(G,n\right)}$$ is constructed to have $${H_{n}=G}$$ and $${H_{i}=0}$$ for $${i\neq n}$$