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Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$

$$0\to U_0 \stackrel{\pa}{\to} U_1\to 0.$$

(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)

This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U_0$ and $U_1$ appropriately we can represent $\pa$ as the identity matrix.

Assume this complex is acyclic and set $n=\dim U_0=\dim U_1$.

The torsion arises when $U_0$ and $U_1$ have additional data attached to them. Assume that $L_0$ and $L_1$ are lattices in $U_0$ and respectively $U_1$ such that $\pa(L_0)\subset L_1$. (These are finitely generated Abelian subgroups that span their respective ambient spaces. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Think of the subgroup $\bZ^n$ of $\bR^n$.)

A $\bZ$-basis $\newcommand{\ue}{\underline{\mathbf{e}}}$ $\newcommand{\uf}{\underline{\mathbf{f}}}$ $\ue_i$ of $L_i$ is also an $\bR$-basis of $U_i$. By choosing $\bZ$-bases $\ue_0$, $\ue_1$ of $L_0$ and respectively $L_1$ we can represent $\pa$ as an $n\times n$ matrix $M(\pa,\ue_0,\ue_1)$.

Observe that if $\uf_0$ and $\uf_1$ are other $\bZ$-bases of $L_0$ resp $L_1$, then

$$|\det M(\pa,\ue_0,\ue_1)|=|\det M(\pa,\uf_0,\uf_1)|=|L_1/\pa L_0|, $$

where $|S|$ denotes the cardinality of the set $S$.

We see that if the associated complex is obtained from a complex of Abelian groups we can associated an invariant, the above determinant, or the order of the quotient $L_1/\pa L_0$. The torsion of this complex is the defined (up to a sign) to be the number

$$\tau(\pa,L_0,L_1):=\pm \frac{1}{\det M(\pa,\ue_0,\ue_1)} =\pm \frac{1}{|L_1/\pa L_0|}. $$

The chain complexes that appear in topology often come equipped with such lattices. Think of simplicial homology with local coefficients, or the chain complex associated to a $CW$-decomposition. You get One such invariant for every triangulation and every choice of local coefficients. For the Reidemeister torsion, the local coefficients are Abelian and correspond to group morphisms $\pi_1\to \mathbb{C}^*$.

It turns out that for smooth manifolds the resulting invariant is independent of the triangulations used to define the torsion.

This is only the beginning of the story and I have omitted many important details. It should help you navigate the first part of my book on torsion where you will find many other descriptions and applications.

Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$

$$0\to U_0 \stackrel{\pa}{\to} U_1\to 0.$$

(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)

This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U_0$ and $U_1$ appropriately we can represent $\pa$ as the identity matrix.

Assume this complex is acyclic and set $n=\dim U_0=\dim U_1$.

The torsion arises when $U_0$ and $U_1$ have additional data attached to them. Assume that $L_0$ and $L_1$ are lattices in $U_0$ and respectively $U_1$ such that $\pa(L_0)\subset L_1$. (These are finitely generated Abelian subgroups that span their respective ambient spaces. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Think of the subgroup $\bZ^n$ of $\bR^n$.)

A $\bZ$-basis $\newcommand{\ue}{\underline{\mathbf{e}}}$ $\newcommand{\uf}{\underline{\mathbf{f}}}$ $\ue_i$ of $L_i$ is also an $\bR$-basis of $U_i$. By choosing $\bZ$-bases $\ue_0$, $\ue_1$ of $L_0$ and respectively $L_1$ we can represent $\pa$ as an $n\times n$ matrix $M(\pa,\ue_0,\ue_1)$.

Observe that if $\uf_0$ and $\uf_1$ are other $\bZ$-bases of $L_0$ resp $L_1$, then

$$|\det M(\pa,\ue_0,\ue_1)|=|\det M(\pa,\uf_0,\uf_1)|=|L_1/\pa L_0|, $$

where $|S|$ denotes the cardinality of the set $S$.

We see that if the associated complex is obtained from a complex of Abelian groups we can associated an invariant, the above determinant, or the order of the quotient $L_1/\pa L_0$. The torsion of this complex is the defined (up to a sign) to be the number

$$\tau(\pa,L_0,L_1):=\pm \frac{1}{\det M(\pa,\ue_0,\ue_1)} =\pm \frac{1}{|L_1/\pa L_0|}. $$

The chain complexes that appear in topology often come equipped with such lattices. Think of simplicial homology with local coefficients, or the chain complex associated to a $CW$-decomposition. You get one such invariant for every triangulation and every choice of local coefficients. For the Reidemeister torsion, the local coefficients are Abelian and correspond to group morphisms $\pi_1\to \mathbb{C}^*$.

It turns out that for smooth manifolds the resulting invariant is independent of the triangulations used to define the torsion.

This is only the beginning of the story and I have omitted many important details. It should help you navigate the first part of my book on torsion where you will find many other descriptions and applications.

Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$

$$0\to U_0 \stackrel{\pa}{\to} U_1\to 0.$$

(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)

This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U_0$ and $U_1$ appropriately we can represent $\pa$ as the identity matrix.

Assume this complex is acyclic and set $n=\dim U_0=\dim U_1$.

The torsion arises when $U_0$ and $U_1$ have additional data attached to them. Assume that $L_0$ and $L_1$ are lattices in $U_0$ and respectively $U_1$ such that $\pa(L_0)\subset L_1$. (These are finitely generated Abelian subgroups that span their respective ambient spaces. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Think of the subgroup $\bZ^n$ of $\bR^n$.)

A $\bZ$-basis $\newcommand{\ue}{\underline{\mathbf{e}}}$ $\newcommand{\uf}{\underline{\mathbf{f}}}$ $\ue_i$ of $L_i$ is also a $\bR$-basis of $U_i$. By choosing $\bZ$-bases $\ue_0$, $\ue_1$ of $L_0$ and respectively $L_1$ we can represent $\pa$ as an $n\times n$ matrix $M(\pa,\ue_0,\ue_1)$.

Observe that if $\uf_0$ and $\uf_1$ are other $\bZ$-bases of $L_0$ resp $L_1$, then

$$|\det M(\pa,\ue_0,\ue_1)|=|\det M(\pa,\uf_0,\uf_1)|=|L_1/\pa L_0|, $$

where $|S|$ denotes the cardinality of the set $S$.

We see that if the associated complex is obtained from a complex of Abelian groups we can associated an invariant, the above determinant, or the order of the quotient $L_1/\pa L_0$. The torsion of this complex is the defined (up to a sign) to be the number

$$\tau(\pa,L_0,L_1):=\pm \frac{1}{\det M(\pa,\ue_0,\ue_1)} =\pm \frac{1}{|L_1/\pa L_0|}. $$

The chain complexes that appear in topology often come equipped with such lattices. Think of simplicial homology, or the chain complex associated to a $CW$-decomposition.

It turns out that for smooth manifolds the resulting invariant is independent of the triangulations used to define the torsion.

This is only the beginning of the story and I have omitted many important details. It should help you navigate the first part of my book on torsion where you will find many other descriptions and applications.

Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$

$$0\to U_0 \stackrel{\pa}{\to} U_1\to 0.$$

(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)

This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U_0$ and $U_1$ appropriately we can represent $\pa$ as the identity matrix.

Assume this complex is acyclic and set $n=\dim U_0=\dim U_1$.

The torsion arises when $U_0$ and $U_1$ have additional data attached to them. Assume that $L_0$ and $L_1$ are lattices in $U_0$ and respectively $U_1$ such that $\pa(L_0)\subset L_1$. (These are finitely generated Abelian subgroups that span their respective ambient spaces. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Think of the subgroup $\bZ^n$ of $\bR^n$.)

A $\bZ$-basis $\newcommand{\ue}{\underline{\mathbf{e}}}$ $\newcommand{\uf}{\underline{\mathbf{f}}}$ $\ue_i$ of $L_i$ is also an $\bR$-basis of $U_i$. By choosing $\bZ$-bases $\ue_0$, $\ue_1$ of $L_0$ and respectively $L_1$ we can represent $\pa$ as an $n\times n$ matrix $M(\pa,\ue_0,\ue_1)$.

Observe that if $\uf_0$ and $\uf_1$ are other $\bZ$-bases of $L_0$ resp $L_1$, then

$$|\det M(\pa,\ue_0,\ue_1)|=|\det M(\pa,\uf_0,\uf_1)|=|L_1/\pa L_0|, $$

where $|S|$ denotes the cardinality of the set $S$.

We see that if the associated complex is obtained from a complex of Abelian groups we can associated an invariant, the above determinant, or the order of the quotient $L_1/\pa L_0$. The torsion of this complex is the defined (up to a sign) to be the number

$$\tau(\pa,L_0,L_1):=\pm \frac{1}{\det M(\pa,\ue_0,\ue_1)} =\pm \frac{1}{|L_1/\pa L_0|}. $$

The chain complexes that appear in topology often come equipped with such lattices. Think of simplicial homology with local coefficients, or the chain complex associated to a $CW$-decomposition. You get One such invariant for every triangulation and every choice of local coefficients. For the Reidemeister torsion, the local coefficients are Abelian and correspond to group morphisms $\pi_1\to \mathbb{C}^*$.

It turns out that for smooth manifolds the resulting invariant is independent of the triangulations used to define the torsion.

This is only the beginning of the story and I have omitted many important details. It should help you navigate the first part of my book on torsion where you will find many other descriptions and applications.

Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$

$$0\to U_0 \stackrel{\pa}{\to} U_1\to 0.$$

(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)

This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U_0$ and $U_1$ appropriately we can represent $\pa$ as the identity matrix.

Assume this complex is acyclic and set $n=\dim U_0=\dim U_1$.

The torsion arises when $U_0$ and $U_1$ have additional data attached to them. Assume that $L_0$ and $L_1$ are lattices in $U_0$ and respectively $U_1$ such that $\pa(L_0)\subset L_1$. (These are finitely generated Abelian subgroups that span their correspondent ambient space. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Think of the subgroup $\bZ^n$ of $\bR^n$.

A $\bZ$-basis $\newcommand{\ue}{\underline{\mathbf{e}}}$ $\newcommand{\uf}{\underline{\mathbf{f}}}$ $\ue_i$ of $L_i$ is also a $\bR$-basis of $U_i$. By choosing $\bZ$-bases $\ue_0$, $\ue_1$ of $L_0$ and respectively $L_1$ we can represent $\pa$ as an $n\times n$ matrix $M(\pa,\ue_0,\ue_1)$.

Observe that if $\uf_0$ and $\uf_1$ are other $\bZ$-bases of $L_0$ resp $L_1$, then

$$\det M(\pa,\ue_0,\ue_1)=\det M(\pa,\uf_0,\uf_1)=|L_1/\pa L_0|, $$

where $|S|$ denotes the cardinality of the set $S$.

We see that if the associated complex is obtained from a complex of Abelian groups we can associated an invariant, the above determinant, or the order of the quotient $L_1/\pa L_0$. The torsion of this complex is the defined to be the number

$$\tau(\pa,L_0,L_1):=\frac{1}{|L_1/\pa L_0|}. $$

The chain complexes that appear in topology often come equipped with such lattices. Think of simplicial homology, or the chain complex associated to a $CW$-decomposition.

It turns out that for smooth manifolds the resulting invariant is independent of the triangulations used to define the torsion.

This is only the beginning of the story and I have omitted many important details. It should help you navigate the first part of my book on torsion where you will find many other descriptions and applications.

Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$

$$0\to U_0 \stackrel{\pa}{\to} U_1\to 0.$$

(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)

This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U_0$ and $U_1$ appropriately we can represent $\pa$ as the identity matrix.

Assume this complex is acyclic and set $n=\dim U_0=\dim U_1$.

The torsion arises when $U_0$ and $U_1$ have additional data attached to them. Assume that $L_0$ and $L_1$ are lattices in $U_0$ and respectively $U_1$ such that $\pa(L_0)\subset L_1$. (These are finitely generated Abelian subgroups that span their respective ambient spaces. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Think of the subgroup $\bZ^n$ of $\bR^n$.)

A $\bZ$-basis $\newcommand{\ue}{\underline{\mathbf{e}}}$ $\newcommand{\uf}{\underline{\mathbf{f}}}$ $\ue_i$ of $L_i$ is also a $\bR$-basis of $U_i$. By choosing $\bZ$-bases $\ue_0$, $\ue_1$ of $L_0$ and respectively $L_1$ we can represent $\pa$ as an $n\times n$ matrix $M(\pa,\ue_0,\ue_1)$.

Observe that if $\uf_0$ and $\uf_1$ are other $\bZ$-bases of $L_0$ resp $L_1$, then

$$|\det M(\pa,\ue_0,\ue_1)|=|\det M(\pa,\uf_0,\uf_1)|=|L_1/\pa L_0|, $$

where $|S|$ denotes the cardinality of the set $S$.

We see that if the associated complex is obtained from a complex of Abelian groups we can associated an invariant, the above determinant, or the order of the quotient $L_1/\pa L_0$. The torsion of this complex is the defined (up to a sign) to be the number

$$\tau(\pa,L_0,L_1):=\pm \frac{1}{\det M(\pa,\ue_0,\ue_1)} =\pm \frac{1}{|L_1/\pa L_0|}. $$

The chain complexes that appear in topology often come equipped with such lattices. Think of simplicial homology, or the chain complex associated to a $CW$-decomposition.

It turns out that for smooth manifolds the resulting invariant is independent of the triangulations used to define the torsion.

This is only the beginning of the story and I have omitted many important details. It should help you navigate the first part of my book on torsion where you will find many other descriptions and applications.