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An **involution** is an endomorphism $\sigma$ whose composition with itself is the identity morphism:

(1)$\sigma \circ \sigma \;=\; id
\,.$

**(involutions are equivalently $\mathbb{Z}/2$-actions)**

Involutions are equivalently the value on the single non-trivial element $\sigma$ in $\mathbb{Z}/2$ of a group action by $\mathbb{Z}/2$.

In this guise, involutions appear throughout representation theory, transformation groups, equivariant homotopy theory, equivariant cohomology, etc.

In particular, involutions, are the defining ingredients of *Real* (with capital “R”!) Whitehead-generalized cohomology theories such as KR-theory, MR-theory, BPR-theory and ER-theory.

**(terminology in algebra)**

An associative algebra equipped with an anti-involution is called a *star-algebra*.

On algebras and other mathematical structures where this makes sense, an **anti-involution** is an anti-homomorphism satisfying (1), instead of a homomorphism (hence an anti-endomorphism and necessarily an anti-automorphism).

**(terminology in topology)**

A topological space equipped with an involutive homeomorphism is sometimes called a *real space* (at least in the context of KR-theory).

In view of Rem. , involutions on topological spaces are equivalently known as *topological G-spaces* for $G =$$\mathbb{Z}/2$. The case of n-spheres with involution is discussed here.

In dependent type theory, there are two different notions of involution.

A **non-coherent involution** on a type $A$ is an element of the type

$\sum_{i:A \to A} \prod_{x:A} i(i(x)) =_{A} x$

A **coherent involution** on a type $A$ is an element of the type

$\sum_{i:A \to A} \sum_{p:\prod_{x:A}i(i(x)) =_{A} x} \prod_{x:A} p(i(x)) =_{_{i(i(i(x))) =_{A} i(x)}} \mathrm{ap}_i(i(i(x)), x, p(x))$

Two involutions $f, g \colon X \to X$ commute if and only if their composition $f g$ is also an involution, as shown by the following manipulations:

$\begin{aligned}
f g
\;=\;
g f
&\;\;\;\;\;\implies\;\;\;\;\;
(f g) (f g)
\;=\;
(f g) (g f)
\;=\;
f (g g) f
\;=\;
f f
\;=\;
id
\\
(f g) (f g)
\;=\;
id
&\;\;\;\;\;\implies\;\;\;\;\;
f g
\;=\;
f
\big(
(f g) (f g)
\big)
g
\;=\;
(f f) (g f) (g g)
\;=\;
g f
\,.
\end{aligned}$

In combinatorics, an important class of involutions are the fixed point free ones: an arbitrary involution on a finite set of cardinality $n$ may be specified by the choice of $k$ elements which are fixed together with a fixed point free involution on the remaining $(n-k)$. The number of fixed point free involutions on a set of $2n$ labelled elements is counted by the double factorial $(2n-1)!! = (2n-1)\cdot (2n-3)\cdot\dots\cdot 3\cdot 1 = \frac{(2n)!}{2^n n!}$, while arbitrary involutions on a set of $n$ labelled elements are counted by OEIS sequence A000085, which also counts the number of Young tableaux with $n$ cells.

An involution on a set $X$ is the same thing as an action of $\mathbb{Z}/2\mathbb{Z}$ on $X$.

More generally, let $(C,\otimes,1)$ be a monoidal category with distributive finite coproducts. The object $2 = 1 + 1$ is equipped with an involution

$not : 2 \to 2$

defined as the copairing $not = [inr,inl]$ of the right and left injections. Moreover, 2 can be given the structure of a monoid in $C$, with unit and multiplication

$false : 1 \to 2 \qquad xor : 2 \otimes 2 \to 2$

defined by $false = inl$ and $xor = [id,not]$ (here we make use of the isomorphism $2 \otimes 2 \cong 2 + 2$ to define $xor$ by copairing). The mapping

$X \mapsto 2 \otimes X \cong X + X$

thus extends to a monad on $C$, sending any object $X$ to the free object equipped with an involution over $X$. Explicitly, the unit $\eta_X : X \to 2\otimes X$ and the multiplication $\mu_X : 2\otimes 2\otimes X \to 2\otimes X$ of the monad are defined by tensoring the unit and the multiplication of the monoid with the identity on $X$, while the involution on $2 \otimes X$ is likewise defined by tensoring the involution on 2 with the identity on $X$.

We then have that involutions in $C$ are precisely the algebras of the monad $(2\otimes-,false\otimes-,xor\otimes-)$. In the forward direction, given an involution $f : X \to X$, we define a monad algebra structure $\alpha : 2\otimes X \to X$ on $X$ by $\alpha = [id,f]$ (again using the isomorphism $2\otimes X \cong X+X$). Conversely, given a monad algebra $\alpha : 2\otimes X \to X$, we can define an endomorphism $f : X \to X$ by $f = \alpha \circ inr$. The monad algebra laws imply that

$\alpha \circ inr \circ \alpha \circ inr = \alpha \circ (2\otimes \alpha) \circ (2\otimes inr) \circ inr = \alpha \circ (xor\otimes id) \circ (2\otimes inr) \circ inr$

and since $xor$ is defined such that $(xor\otimes id) \circ (2\otimes inr) \circ inr = id$, we derive that $\alpha \circ inr$ is an involution.

Discussion in combinatorics:

- Philippe Flajolet and Robert Sedgewick,
*Analytic Combinatorics*, CUP, 2009. (author pdf)

Discussion in differential topology:

- Santiago López de Medrano,
*Involutions on Manifolds*, Ergebnisse der Mathematik und ihrer Grenzgebiete**59**, Springer 1971 (doi:10.1007/978-3-642-65012-3)

Last revised on August 21, 2024 at 02:12:38. See the history of this page for a list of all contributions to it.