Homotopy Theory via Multi-Valued Functions

1. Introduction

Classical homotopy theory studies continuous maps

$$f:X⟶Y$$

and identifies maps which can be continuously deformed into one another. The resulting notion of homotopy provides one of the fundamental equivalence relations in algebraic topology.

There are, however, many situations in which a point does not have a unique image. Instead, a point $x\in X$ may determine a nonempty subset of Y. This leads naturally to the concept of a multi-valued, or set-valued, mapping

$$F:X⊸Y,x⟼F\left(x\right)\subseteq Y.$$

The theory of multivalued mappings has developed extensively in topology, nonlinear analysis, fixed point theory, and differential inclusions. A systematic topological treatment was developed by Górniewicz, including upper and lower semicontinuity, selectors, Vietoris maps, admissible mappings, homotopy, and Lefschetz-type methods [2,3]. A later survey of multivalued fixed point theory also emphasizes the role of admissible maps and Vietoris maps in transferring homological information to the multivalued setting [1]. The purpose of this paper is to develop the basic idea suggested by the title:

What happens to homotopy theory if continuous functions are replaced by suitable multi-valued functions?

There are several possible answers. One possibility is to pass to the hyperspace of subsets of Y and regard F as an ordinary map

$$F:X⟶\mathcal{K}\left(Y\right),$$

where $\mathcal{K}\left(Y\right)$ denotes an appropriate hyperspace. Another, more algebraic, possibility is to represent F by a diagram

$$X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y$$

and use the homological properties of p. The latter approach is particularly useful when p is a Vietoris map. In that case the Vietoris theorem gives an isomorphism on suitable homology groups [2,3,5].

We shall use the notation

$$F:X⊸Y$$

for a multi-valued mapping, reserving

$$f:X\to Y$$

for an ordinary single-valued map.

The paper is organized as follows. Section 2 recalls the basic topological notions concerning multi-valued maps. Section 3 discusses Vietoris representations and admissible mappings. Section 4 introduces multi-valued homotopies. Section 5 studies selectors. Section 6 establishes basic homotopy properties. Section 7 describes induced homology maps. Section 8 discusses products and compositions. Section 9 introduces a multi-valued homotopy category. Section 10 gives examples, while Section 11 explains connections with fixed point theory. Section 12 contains concluding remarks and several directions for further research.

2. Multi-Valued Functions

We begin with the basic definition.

Definition 2.1.

Let X and Y be topological spaces. A multi-valued function from X to Y is a mapping

$$F:X⊸Y$$

such that

$$F\left(x\right)\subseteq Y$$

is nonempty for every $x\in X$.

Thus, unlike an ordinary function, F is allowed to assign several points of Y to a single point of X.

Example 2.2.

Let

$$F:[0,1]⊸\mathbb{R}$$

be defined by

$$F\left(x\right)=\{-x,x\}.$$

Then F is a two-valued function. At $x=0$ the value is

$$F\left(0\right)=\left\{0\right\}.$$

Definition 2.3.

A multi-valued function $F:X⊸Y$ is called compact-valued if $F\left(x\right)$ is a nonempty compact subset of Y for every $x\in X$.

We write

$$\mathcal{K}\left(Y\right)=\{A\subseteq Y:A\ne ⌀,A\mathrm{is}\mathrm{compact}\}.$$

Hence a compact-valued mapping may be regarded formally as a map

$$F:X⟶\mathcal{K}\left(Y\right).$$

This observation is one of the basic links between multivalued analysis and hyperspace topology. Hyperspaces and their topology play an important role in the systematic treatment of multivalued mappings [3].

3. Upper and Lower Semicontinuity

Continuity for multi-valued maps has several possible formulations. The most important ones for our purposes are upper and lower semicontinuity.

Definition 3.1.

Let $F:X⊸Y$ be a multi-valued function.

We say that F is upper semicontinuous at $x_0\in X$ if, for every open set $V\subseteq Y$ satisfying

$$F\left(x_0\right)\subseteq V,$$

there exists an open neighborhood U of $x_0$ such that

$$F\left(x\right)\subseteq V$$

for every $x\in U$.

The map is upper semicontinuous, abbreviated usc, if it is upper semicontinuous at every point.

Definition 3.2.

The map $F:X⊸Y$ is lower semicontinuous at $x_0$ if, for every open set $V\subseteq Y$ such that

$$F\left(x_0\right)\cap V\ne ⌀,$$

there is an open neighborhood U of $x_0$ such that

$$F\left(x\right)\cap V\ne ⌀$$

for every $x\in U$.

The distinction between upper and lower semicontinuity is fundamental in selection theory and in the general theory of multivalued mappings [3].

Proposition 3.3.

Let Y be compact Hausdorff and let

$$F:X⊸Y$$

be compact-valued. Then F is upper semicontinuous if and only if its graph

$$Gr\left(F\right)=\left\{\right(x,y)\in X\times Y:y\in F(x\left)\right\}$$

is closed in $X\times Y$.

Proof. 

This is a standard graph characterization of upper semicontinuity for compact-valued mappings into compact Hausdorff spaces. The proof follows from the fact that compact subsets of a Hausdorff space are closed and from the tube lemma. □

4. Vietoris Maps and Admissible Mappings

The passage from ordinary maps to multi-valued maps introduces a technical problem: there is generally no canonical map on homology associated directly with a set-valued function. Vietoris maps provide one of the fundamental mechanisms for overcoming this difficulty.

Definition 4.1.

A continuous surjection

$$p:\mathsf{\Gamma }⟶X$$

is called a Vietoris map if it is proper and every fiber

$$p^{-1}\left(x\right)$$

is acyclic.

Here, acyclicity may be formulated in terms of reduced homology:

$${\tilde{H}}_q\left(p^{-1}\left(x\right)\right)=0$$

for all $q\ge 0$.

The classical Vietoris theorem states, under appropriate hypotheses, that a Vietoris map induces an isomorphism in the relevant homology theory [2,3,5].

Theorem 4.2

(Vietoris). Let

$$p:\mathsf{\Gamma }⟶X$$

be a Vietoris map. Then

$$p_{*}:H_{*}\left(\mathsf{\Gamma }\right)⟶H_{*}\left(X\right)$$

is an isomorphism for the homology theory under consideration.

This theorem is one of the central tools in the topological theory of multi-valued maps. It permits us to replace a complicated multi-valued mapping by a diagram involving an ordinary map whose homological behavior is well controlled [1].

Definition 4.3.

Let

$$F:X⊸Y$$

be a multi-valued mapping. We call F admissible if there exist a topological space $\mathsf{\Gamma }$ and continuous maps

$$X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y$$

such that p is a Vietoris map and

$$F\left(x\right)=q\left(p^{-1}\left(x\right)\right)$$

for every $x\in X$.

The pair

$$(p,q)$$

will be called a selected pair or Vietoris representation of F.

This representation is central in Górniewicz’s approach to multivalued mappings and allows homological constructions to be associated with admissible maps [2,3].

5. Selectors

Selectors provide a direct bridge between ordinary and multi-valued functions.

Definition 5.1.

Let

$$F:X⊸Y.$$

A continuous map

$$f:X⟶Y$$

is called a continuous selector of F if

$$f\left(x\right)\in F\left(x\right)$$

for every $x\in X$.

We write

$$f\subseteq F$$

when f is a selector of F.

Example 5.2.

Let

$$F\left(x\right)=[-x,x],x\in [0,1].$$

Then

$$f\left(x\right)=0$$

is a continuous selector of F. Another selector is

$$g\left(x\right)=x.$$

Selection theory gives conditions under which continuous selectors exist. Michael’s selection theorem is one of the foundational results in this direction and is treated in the standard theory of multivalued mappings [3,4].

Remark 5.3.

The existence of a selector is not automatic. Consequently, a homotopy theory based exclusively on selectors would exclude many interesting multi-valued maps. The admissible-map approach is therefore more flexible.

6. Multi-Valued Homotopy

We now introduce the principal concept of this paper.

Definition 6.1.

Let

$$F,G:X⊸Y$$

be multi-valued mappings. A multi-valued homotopy from F to G is a multi-valued map

$$H:X\times I⊸Y$$

such that

$$H(x,0)=F\left(x\right)$$

and

$$H(x,1)=G\left(x\right)$$

for every $x\in X$.

We write

$$F{\simeq }_{\mathrm{mv}}G.$$

The notation ${\simeq }_{\mathrm{mv}}$ distinguishes multi-valued homotopy from ordinary homotopy.

Remark 6.2.

For admissible mappings, this definition agrees with the standard homotopy concept used in multivalued topology, where the homotopy is required to remain within the class of admissible maps [1,2,3].

Theorem 6.3.

Let $F,G:X⊸Y$ be admissible multi-valued maps. If

$$F{\simeq }_{\mathrm{mv}}G,$$

then the homological information associated with F and G agrees in the sense determined by their Vietoris representations.

Proof. 

Choose admissible representations

$$X\stackrel{p_F}{\leftarrow }{\mathsf{\Gamma }}_F\stackrel{q_F}{\to }Y$$

and

$$X\stackrel{p_G}{\leftarrow }{\mathsf{\Gamma }}_G\stackrel{q_G}{\to }Y.$$

By the Vietoris theorem, the maps induced by $p_F$ and $p_G$ are isomorphisms on the relevant homology groups. A multi-valued homotopy between F and G gives an admissible representation over $X\times I$. Restriction to the two ends of the interval produces representations of F and G. Since the inclusions

$$X\times \left\{0\right\}\hookrightarrow X\times I$$

and

$$X\times \left\{1\right\}\hookrightarrow X\times I$$

are homotopic, the induced homological maps agree up to the canonical identifications. Hence the homological invariants associated with F and G coincide. □

7. Elementary Properties of Multi-Valued Homotopy

The multi-valued homotopy relation has the expected formal properties.

Theorem 7.1.

On the class of admissible maps from X to Y, the relation

$${\simeq }_{\mathrm{mv}}$$

is an equivalence relation.

Proof. 

We verify the three properties.

Reflexivity. For an admissible map $F:X⊸Y$, define

$$H(x,t)=F\left(x\right).$$

Then

$$H(x,0)=F\left(x\right)=H(x,1),$$

so

$$F{\simeq }_{\mathrm{mv}}F.$$

Symmetry. Suppose

$$F{\simeq }_{\mathrm{mv}}G$$

through H. Define

$$\overline{H}(x,t)=H(x,1-t).$$

Then

$$\overline{H}(x,0)=G\left(x\right),\overline{H}(x,1)=F\left(x\right).$$

Thus

$$G{\simeq }_{\mathrm{mv}}F.$$

Transitivity. Suppose

$$F{\simeq }_{\mathrm{mv}}G$$

through $H_1$ and

$$G{\simeq }_{\mathrm{mv}}K$$

through $H_2$. Define

$$H(x,t)=\left\{\begin{array}{cc}{H}_1(x,2t),\hfill & 0\le t\le {\displaystyle \frac{1}{2}},\hfill \\ H_2(x,2t-1),\hfill & {\displaystyle \frac{1}{2}}\le t\le 1.\hfill \end{array}\right.$$

The two pieces agree at $t=1/2$, because

$$H_1(x,1)=G\left(x\right)=H_2(x,0).$$

Hence

$$F{\simeq }_{\mathrm{mv}}K.$$

□

8. Composition of Multi-Valued Maps

If

$$F:X⊸Y$$

and

$$G:Y⊸Z,$$

their composition is naturally defined by

$$(G\circ F)\left(x\right)=\bigcup _{y\in F\left(x\right)}G\left(y\right).$$

Definition 8.1.

The composition of multi-valued maps

$$F:X⊸Y,G:Y⊸Z$$

is the multi-valued map

$$G\circ F:X⊸Z$$

given by

$$(G\circ F)\left(x\right)=\bigcup _{y\in F\left(x\right)}G\left(y\right).$$

Proposition 8.2.

Multi-valued composition is associative whenever all compositions are well-defined.

Proof. 

For $x\in X$,

$$((H\circ G)\circ F)\left(x\right)=\bigcup _{y\in F\left(x\right)}\bigcup _{z\in G\left(y\right)}H\left(z\right).$$

On the other hand,

$$(H\circ (G\circ F))\left(x\right)=\bigcup _{z\in (G\circ F)\left(x\right)}H\left(z\right)=\bigcup _{y\in F\left(x\right)}\bigcup _{z\in G\left(y\right)}H\left(z\right).$$

Therefore the two sets are equal. □

Theorem 8.3.

Suppose

$$F_0{\simeq }_{\mathrm{mv}}{F}_1:X⊸Y$$

and

$$G_0{\simeq }_{\mathrm{mv}}{G}_1:Y⊸Z.$$

Under hypotheses guaranteeing that the compositions remain admissible, one has

$$G_0\circ F_0{\simeq }_{\mathrm{mv}}{G}_1\circ F_1.$$

Proof. 

Let $H_F$ be a multi-valued homotopy from $F_0$ to $F_1$ and let $H_G$ be one from $G_0$ to $G_1$. First deform $F_0$ into $F_1$ while keeping $G_0$ fixed, and then deform $G_0$ into $G_1$ while keeping $F_1$ fixed. Concatenating these two homotopies gives the required homotopy. □

9. Homotopy of Selectors

The selector viewpoint provides a particularly transparent connection with ordinary homotopy theory.

Proposition 9.1.

Let

$$F:X⊸Y$$

have two continuous selectors

$$f,g:X⟶Y.$$

Suppose there exists a continuous family of selectors

$$h_t:X⟶Y,t\in I,$$

such that

$$h_0=f,h_1=g.$$

Then

$$f\simeq g$$

in the ordinary sense.

Proof. 

Define

$$H:X\times I⟶Y,H(x,t)=h_t\left(x\right).$$

Continuity gives an ordinary homotopy from f to g. □

More generally, a multi-valued homotopy may contain a family of ordinary homotopies between different selectors. This suggests a hierarchy:

$$\mathrm{ordinary}\mathrm{homotopy}\subseteq \mathrm{selector}\mathrm{homotopy}\subseteq \mathrm{multi}\text{-}\mathrm{valued}\mathrm{homotopy}.$$

Remark 9.2.

The inclusions above should not be interpreted as equality of categories. A multi-valued map may possess no continuous selector at all, while still being admissible and participating in a well-defined multi-valued homotopy theory.

10. Induced Maps on Homology

We next explain how homological information can be attached to an admissible multi-valued map.

Let

$$F:X⊸Y$$

be admissible and choose a representation

$$X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y.$$

Since p is a Vietoris map,

$$p_{*}:H_{*}\left(\mathsf{\Gamma }\right)⟶H_{*}\left(X\right)$$

is an isomorphism. We may therefore define

$$F_{*}=q_{*}\circ p_{*}^{-1}.$$

Definition 10.1.

For an admissible map

$$F:X⊸Y$$

represented by

$$X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y,$$

its induced homological morphism is

$$F_{*}=q_{*}\circ p_{*}^{-1}.$$

This construction is closely related to the standard treatment of admissible mappings in multivalued fixed point theory [2,3].

Theorem 10.2.

If

$$F{\simeq }_{\mathrm{mv}}G:X⊸Y,$$

then the corresponding induced homological morphisms are equal, whenever the chosen homological framework makes the construction single-valued.

Proof. 

A multi-valued homotopy produces a representation over $X\times I$. The restrictions at $t=0$ and $t=1$ represent F and G. Homotopy invariance of homology implies equality of the induced maps after the canonical identifications. □

Thus the functorial philosophy of algebraic topology survives in the multi-valued setting, provided the class of maps is chosen carefully.

11. Products and Multi-Valued Homotopies

Products behave naturally in the multi-valued framework.

Let

$$F:X⊸Y$$

and

$$G:Z⊸W.$$

Define

$$F\times G:X\times Z⊸Y\times W$$

by

$$(F\times G)(x,z)=F\left(x\right)\times G\left(z\right).$$

Proposition 11.1.

If F and G are compact-valued, then $F\times G$ is compact-valued.

Proof. 

The product of two nonempty compact subsets is nonempty and compact. Hence

$$F\left(x\right)\times G\left(z\right)$$

is compact for every $(x,z)\in X\times Z$. □

Theorem 11.2.

Suppose

$$F_0{\simeq }_{\mathrm{mv}}{F}_1$$

and

$$G_0{\simeq }_{\mathrm{mv}}{G}_1.$$

Then

$$F_0\times G_0{\simeq }_{\mathrm{mv}}{F}_1\times G_1.$$

Proof. 

Let $H_F$ and $H_G$ be the corresponding homotopies. Define

$$H((x,z),t)=H_F(x,t)\times H_G(z,t).$$

Then

$$H((x,z),0)=F_0\left(x\right)\times G_0\left(z\right)$$

and

$$H((x,z),1)=F_1\left(x\right)\times G_1\left(z\right).$$

Under the usual stability assumptions on the chosen class of multi-valued maps, this is an admissible multi-valued homotopy. □

12. Multi-Valued Homotopy Equivalence

We now formulate the analogue of homotopy equivalence.

Definition 12.1.

Let X and Y be topological spaces. We say that X and Y are multi-valuedly homotopy equivalent if there exist admissible maps

$$F:X⊸Y,G:Y⊸X$$

such that

$$G\circ F{\simeq }_{\mathrm{mv}}{id}_X$$

and

$$F\circ G{\simeq }_{\mathrm{mv}}{id}_Y.$$

We write

$$X{\simeq }_{\mathrm{mv}}Y.$$

This definition permits an object to be connected to another object without requiring either direction to be represented by a unique ordinary function.

Theorem 12.2.

Multi-valued homotopy equivalence is an equivalence relation on the class of spaces for which the required admissible compositions exist.

Proof. 

Reflexivity follows from the identity map.

For symmetry, if $F:X⊸Y$ and $G:Y⊸X$ establish a multi-valued homotopy equivalence, then the same pair with the order reversed establishes

$$Y{\simeq }_{\mathrm{mv}}X.$$

For transitivity, suppose

$$X{\simeq }_{\mathrm{mv}}Y$$

through $F,G$ and

$$Y{\simeq }_{\mathrm{mv}}Z$$

through $P,Q$. Consider

$$P\circ F:X⊸Z$$

and

$$G\circ Q:Z⊸X.$$

Associativity of composition and the homotopies defining the two equivalences give the required homotopies to the identity maps. □

13. The Multi-Valued Homotopy Category

The preceding constructions suggest a categorical formulation.

Let

$$\mathbf{MVTop}$$

denote a category whose objects are suitable topological spaces and whose morphisms are admissible multi-valued mappings.

We then define the multi-valued homotopy category

$$\mathbf{HoMVTop}$$

as follows.

Definition 13.1.

The objects of $\mathbf{HoMVTop}$ are the objects of $\mathbf{MVTop}$. A morphism from X to Y is a multi-valued homotopy class

$$\left[F\right]$$

of admissible maps

$$F:X⊸Y.$$

Composition is defined by

$$\left[G\right]\circ \left[F\right]=[G\circ F].$$

Theorem 13.2.

The category $\mathbf{HoMVTop}$ is well-defined provided admissible maps are closed under composition and multi-valued homotopy is compatible with composition.

Proof. 

Suppose

$$F{\simeq }_{\mathrm{mv}}{F}^{\prime }$$

and

$$G{\simeq }_{\mathrm{mv}}{G}^{\prime }.$$

By compatibility of homotopy with composition,

$$G\circ F{\simeq }_{\mathrm{mv}}{G}^{\prime }\circ F^{\prime }.$$

Therefore the homotopy class of $G\circ F$ depends only on the homotopy classes of F and G. Associativity follows from associativity of multi-valued composition. Identity morphisms are represented by the ordinary identity maps. □

This category provides a natural environment in which classical homotopy theory can be extended from functions to set-valued mappings.

14. Examples

Example 14.1

(A constant multi-valued map). Let $A\subseteq Y$ be nonempty and define

$$F:X⊸Y,F\left(x\right)=A$$

for every $x\in X$.

If A is contractible, then the map is naturally associated with the homotopy type of a point.

Example 14.2

(A two-valued map). Consider

$$F:S^1⊸S^1$$

defined by

$$F\left(z\right)=\{z,-z\}.$$

The map has two values at every point. It can be interpreted as a finite-valued correspondence rather than a classical function. Questions concerning its homotopy behavior may be studied through selectors, hyperspaces, or admissible representations.

Example 14.3

(Interval-valued maps). Let

$$F:X⊸\mathbb{R}$$

be defined by

$$F\left(x\right)=\left[f\right(x),g(x\left)\right]$$

where

$$f,g:X⟶\mathbb{R}$$

are continuous and

$$f\left(x\right)\le g\left(x\right).$$

Then

$$x⟼\left[f\right(x),g(x\left)\right]$$

is a natural compact-valued multi-valued map. The functions f and g are selectors.

Example 14.4

(A contractible-valued correspondence). Suppose

$$F:X⊸Y$$

has contractible compact values and is upper semicontinuous. Under additional hypotheses ensuring admissibility, the map can often be handled by Vietoris methods. This is one of the reasons acyclic and contractible values are important in multivalued topology [1,3].

15. Relation with Fixed Point Theory

Multi-valued homotopy theory is closely connected with multivalued fixed point theory.

For

$$F:X⊸X$$

define

$$Fix\left(F\right)=\{x\in X:x\in F(x\left)\right\}.$$

Definition 15.1.

A point $x\in X$ is a fixed point of $F:X⊸X$ if

$$x\in F\left(x\right).$$

The classical theory of admissible multi-valued mappings uses homological constructions to establish fixed point and Lefschetz-type results [2,3]. Modern surveys likewise emphasize the role of Vietoris maps, admissible maps, and homotopy in multivalued fixed point theory [1].

Theorem 15.2

(Homotopy invariance principle). Let $F,G:X⊸X$ be admissible maps such that

$$F{\simeq }_{\mathrm{mv}}G.$$

Suppose that a Lefschetz-type invariant

$$\mathsf{\Lambda }\left(F\right)$$

is defined and is invariant under multi-valued homotopy. Then

$$\mathsf{\Lambda }\left(F\right)=\mathsf{\Lambda }\left(G\right).$$

Consequently, any fixed point theorem based only on the nonvanishing of Λ applies simultaneously to F and G.

Remark 15.3.

The theorem is deliberately formulated at an abstract level. The precise definition of $\mathsf{\Lambda }\left(F\right)$ depends on the selected homology theory and the class of admissible maps. In the classical theory, Lefschetz-type constructions are developed for admissible mappings and their generalizations [3].

16. A Homotopy Invariant for Multi-Valued Maps

Let

$$F:X⊸Y$$

be admissible. Choose a representation

$$X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y.$$

Assume that the corresponding homology map

$$F_{*}=q_{*}{p}_{*}^{-1}$$

is well-defined.

We may define the multi-valued homological type of F by

$$\mathfrak{H}\left(F\right)=\left\{q_{*}{p}_{*}^{-1}\right\}.$$

Proposition 16.1.

If

$$F{\simeq }_{\mathrm{mv}}G,$$

then

$$\mathfrak{H}\left(F\right)=\mathfrak{H}\left(G\right)$$

whenever the homological realization is independent of the selected Vietoris representation.

Proof. 

This follows directly from homotopy invariance of homology applied to a Vietoris representation of the multi-valued homotopy. □

This suggests that one can define higher algebraic invariants of a multi-valued mapping by applying ordinary algebraic topology to a suitable Vietoris representation.

17. Connection with Classical Homotopy Theory

The theory developed here contains ordinary homotopy theory as a special case.

Every continuous function

$$f:X⟶Y$$

defines a multi-valued map by

$$F_f\left(x\right)=\left\{f\left(x\right)\right\}.$$

Proposition 17.1.

The correspondence

$$f⟼F_f$$

embeds ordinary continuous maps into the class of multi-valued maps.

Proof. 

For every $x\in X$,

$$F_f\left(x\right)=\left\{f\left(x\right)\right\}$$

is a nonempty compact subset of Y. Thus $F_f$ is a multi-valued map. Ordinary composition agrees with multi-valued composition:

$$F_g\circ F_f=F_{g\circ f}.$$

□

Theorem 17.2.

If

$$f,g:X⟶Y$$

are ordinary continuous maps, then

$$f\simeq g$$

in the classical sense if and only if

$$F_f{\simeq }_{\mathrm{mv}}{F}_g$$

through a single-valued homotopy.

Proof. 

A classical homotopy

$$H:X\times I⟶Y$$

defines the multi-valued homotopy

$$\tilde{H}(x,t)=\left\{H(x,t)\right\}.$$

The converse follows immediately when the multi-valued homotopy is single-valued at every point. □

Thus the proposed framework is genuinely an extension rather than a replacement of ordinary homotopy theory.

18. Homotopy Groups in the Multi-Valued Setting

The preceding discussion naturally raises the question of whether one can define homotopy groups using multi-valued maps.

Let $(X,x_0)$ be a pointed topological space. A multi-valued loop may be viewed as an admissible map

$$F:(I,\{0,1\})⊸(X,x_0)$$

with suitable endpoint conditions.

One possible definition is to consider homotopy classes of admissible multi-valued maps

$$F:(I^n,\partial I^n)⊸(X,x_0).$$

Definition 18.1.

The set

$${\pi }_n^{\mathrm{mv}}(X,x_0)$$

is defined formally as the set of multi-valued homotopy classes of admissible maps

$$(I^n,\partial I^n)⊸(X,x_0).$$

For this definition to produce a group, additional assumptions are needed, particularly concerning composition, admissibility, and concatenation.

Remark 18.2.

The construction above should be regarded as a starting point rather than a universal definition. Different choices of admissible maps, values, endpoint conditions, and homotopy relations can lead to different versions of multi-valued homotopy groups.

A particularly interesting problem is to identify hypotheses under which

$${\pi }_n^{\mathrm{mv}}(X,x_0)$$

agrees with the classical homotopy group

$${\pi }_n(X,x_0).$$

19. Conclusion

We have presented a framework for studying homotopy theory through multi-valued functions. The fundamental idea is to replace a function

$$f:X\to Y$$

by a correspondence

$$F:X⊸Y$$

and then impose enough topological structure to recover meaningful homotopy-theoretic information.

The principal tools are compact-valued upper semicontinuous mappings, Vietoris maps, admissible representations, selectors, multi-valued homotopies, and induced homology morphisms. The Vietoris theorem provides the crucial bridge between the ordinary and multi-valued settings, while the theory of admissible mappings gives a flexible framework for composition and homotopy.

The construction contains ordinary homotopy theory as the special case of singleton-valued maps. At the same time, it permits genuinely multi-valued phenomena which cannot be represented by a single continuous function.

The resulting perspective suggests that multi-valued topology should not be regarded merely as an auxiliary technique for fixed point theory. It can instead be viewed as a possible extension of the homotopy-theoretic language itself. In particular, the category

$$\mathbf{HoMVTop}$$

and the proposed multi-valued homotopy groups provide natural starting points for further investigation.