In mathematics, **Out( F_{n})** is the outer automorphism group of a free group on

Out(*F _{n}*) acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Teichmüller space for a bouquet of circles.

A point of the outer space is essentially an -graph *X* homotopy equivalent to a bouquet of *n* circles together with a certain choice of a free homotopy class of a homotopy equivalence from *X* to the bouquet of *n* circles. An -graph is just a weighted graph with weights in . The sum of all weights should be 1 and all weights should be positive. To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least 3.

A more descriptive view avoiding the homotopy equivalence *f* is the following. We may fix an identification of the fundamental group of the bouquet of *n* circles with the free group in *n* variables. Furthermore, we may choose a maximal tree in *X* and choose for each remaining edge a direction. We will now assign to each remaining edge *e* a word in in the following way. Consider the closed path starting with *e* and then going back to the origin of *e* in the maximal tree. Composing this path with *f* we get a closed path in a bouquet of *n* circles and hence an element in its fundamental group . This element is not well defined; if we change *f* by a free homotopy we obtain another element. It turns out, that those two elements are conjugate to each other, and hence we can choose the unique cyclically reduced element in this conjugacy class. It is possible to reconstruct the free homotopy type of *f* from these data. This view has the advantage, that it avoids the extra choice of *f* and has the disadvantage that additional ambiguity arises, because one has to choose a maximal tree and an orientation of the remaining edges.

The operation of Out(*F _{n}*) on the outer space is defined as follows. Every automorphism

Every point in the outer space determines a unique length function . A word in determines via the chosen homotopy equivalence a closed path in *X*. The length of the word is then the minimal length of a path in the free homotopy class of that closed path. Such a length function is constant on each conjugacy class. The assignment defines an embedding of the outer space to some infinite dimensional projective space.

In the second model an open simplex is given by all those -graphs, which have combinatorically the same underlying graph and the same edges are labeled with the same words (only the length of the edges may differ). The boundary simplices of such a simplex consists of all graphs, that arise from this graph by collapsing an edge. If that edge is a loop it cannot be collapsed without changing the homotopy type of the graph. Hence there is no boundary simplex. So one can think about the outer space as a simplicial complex with some simplices removed. It is easy to verify, that the action of is simplicial and has finite isotropy groups.

The abelianization map induces a homomorphism from to the general linear group , the latter being the automorphism group of . This map is onto, making a group extension,

- .

The kernel is the Torelli group of .

In the case , the map is an isomorphism.

Because is the fundamental group of a bouquet of *n* circles, can be described topologically as the mapping class group of a bouquet of *n* circles (in the homotopy category), in analogy to the mapping class group of a closed surface which is isomorphic to the outer automorphism group of the fundamental group of that surface.

In the mathematical field of algebraic topology, the **fundamental group** of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.

In mathematics, especially in category theory and homotopy theory, a **groupoid** generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In mathematics, **homology** is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In topology, a branch of mathematics, two continuous functions from one topological space to another are called **homotopic** if one can be "continuously deformed" into the other, such a deformation being called a **homotopy** between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In mathematics, specifically algebraic topology, a **covering map** is a continuous function from a topological space to a topological space such that each point in has an open neighborhood **evenly covered** by . In this case, is called a **covering space** and the **base space** of the covering projection. The definition implies that every covering map is a local homeomorphism.

In mathematics, **homotopy groups** are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or *holes*, of a topological space.

A **CW complex** is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. The *C* stands for "closure-finite", and the *W* for "weak" topology. A CW complex can be defined inductively.

In algebraic geometry, **motives** is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its **Whitehead torsion** which is an element in the **Whitehead group**. These concepts are named after the mathematician J. H. C. Whitehead.

In topology, especially algebraic topology, the **cone****of a topological space** is the quotient space:

In mathematics, in the subfield of geometric topology, the **mapping class group** is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.

In mathematics, specifically in homotopy theory, a **classifying space***BG* of a topological group *G* is the quotient of a weakly contractible space *EG* by a proper free action of *G*. It has the property that any *G* principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle *EG* → *BG*. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy.

In the mathematical subject of geometric group theory, the **Culler–Vogtmann Outer space** or just **Outer space** of a free group *F*_{n} is a topological space consisting of the so-called "marked metric graph structures" of volume 1 on *F*_{n}. The Outer space, denoted *X*_{n} or *CV*_{n}, comes equipped with a natural action of the group of outer automorphisms Out(*F*_{n}) of *F*_{n}. The Outer space was introduced in a 1986 paper, of Marc Culler and Karen Vogtmann and it serves as a free group analog of the Teichmüller space of a hyperbolic surface. Outer space is used to study homology and cohomology groups of Out(*F*_{n}) and to obtain information about algebraic, geometric and dynamical properties of Out(*F*_{n}), of its subgroups and individual outer automorphisms of *F*_{n}. The space *X*_{n} can also be thought of as the set of *F*_{n}-equivariant isometry types of minimal free discrete isometric actions of *F*_{n} on *F*_{n} on **R**-trees*T* such that the quotient metric graph *T*/*F*_{n} has volume 1.

In the mathematical subject of geometric group theory, a **train track map** is a continuous map *f* from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations. This map sends vertices to vertices and edges to nontrivial edge-paths with the property that for every edge *e* of the graph and for every positive integer *n* the path *f ^{n}*(

In mathematics, a **Δ-set***S*, often called a **semi-simplicial set**, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.

**Karen Vogtmann** is an American mathematician working primarily in the area of geometric group theory. She is known for having introduced, in a 1986 paper with Marc Culler, an object now known as the Culler–Vogtmann Outer space. The Outer space is a free group analog of the Teichmüller space of a Riemann surface and is particularly useful in the study of the group of outer automorphisms of the free group on *n* generators, Out(*F*_{n}). Vogtmann is a professor of mathematics at Cornell University and The University of Warwick.

In mathematics, and more precisely in topology, the **mapping class group** of a surface, sometimes called the **modular group** or **Teichmüller modular group**, is the group of homeomorphisms of the surface viewed up to continuous deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.

This is a glossary of properties and concepts in algebraic topology in mathematics.

In mathematics, the **free factor complex** is a free group counterpart of the notion of the curve complex of a finite type surface. The free factor complex was originally introduced in a 1998 paper of Hatcher and Vogtmann. Like the curve complex, the free factor complex is known to be Gromov-hyperbolic. The free factor complex plays a significant role in the study of large-scale geometry of .

**Whitehead's algorithm** is a mathematical algorithm in group theory for solving the automorphic equivalence problem in the finite rank free group *F _{n}*. The algorithm is based on a classic 1936 paper of J. H. C. Whitehead. It is still unknown if Whitehead's algorithm has polynomial time complexity.

- Culler, Marc; Vogtmann, Karen (1986). "Moduli of graphs and automorphisms of free groups" (PDF).
*Inventiones Mathematicae*.**84**(1): 91–119. doi:10.1007/BF01388734. MR 0830040. - Vogtmann, Karen (2002). "Automorphisms of free groups and outer space" (PDF).
*Geometriae Dedicata*.**94**: 1–31. doi:10.1023/A:1020973910646. MR 1950871. - Vogtmann, Karen (2008), "What is … outer space?" (PDF),
*Notices of the American Mathematical Society*,**55**(7): 784–786, MR 2436509

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