of maps and are key concepts in algebraic topology. They provide a way to compare continuous functions and topological spaces, allowing us to group similar objects and study their shared properties.
These ideas form the foundation for understanding the fundamental group and other homotopy groups. By focusing on continuous deformations, we can explore the essential structure of spaces and maps, leading to powerful classification tools in topology.
Homotopy of Maps
Definition and Notation
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A homotopy between two continuous maps f,g:X→Y is a H:X×[0,1]→Y such that H(x,0)=f(x) and H(x,1)=g(x) for all x∈X
The interval [0,1] represents the parameter space for the homotopy
The homotopy continuously deforms the map f into the map g as the parameter varies from 0 to 1
If a homotopy exists between f and g, then f and g are said to be homotopic, denoted by f≃g
The set of homotopy classes of maps from X to Y is denoted by [X,Y]
Properties of Homotopy
The homotopy relation is an equivalence relation on the set of continuous maps from X to Y
Reflexivity: Every map is homotopic to itself via the constant homotopy H(x,t)=f(x)
Symmetry: If f≃g, then g≃f by reversing the homotopy H(x,t)=H(x,1−t)
Transitivity: If f≃g and g≃h, then f≃h by concatenating the homotopies
induce the same homomorphisms on homotopy groups and homology groups
If f≃g, then f∗:πn(X)→πn(Y) and g∗:πn(X)→πn(Y) are equal for all n≥0
If f≃g, then f∗:Hn(X)→Hn(Y) and g∗:Hn(X)→Hn(Y) are equal for all n≥0
Homotopic Maps and Equivalences
Homotopic Maps
Two continuous maps f,g:X→Y are homotopic if there exists a homotopy H:X×[0,1]→Y between them
Example: The identity map idX:X→X and any constant map c:X→X are homotopic via H(x,t)=(1−t)x+tc(x)
Homotopic maps share many topological properties, such as inducing the same homomorphisms on homotopy and homology groups
Homotopy Equivalences
A homotopy equivalence between two topological spaces X and Y is a pair of continuous maps f:X→Y and g:Y→X such that g∘f≃idX and f∘g≃idY, where idX and idY are the identity maps on X and Y, respectively
The maps f and g are called homotopy equivalences or homotopy inverses of each other
If a homotopy equivalence exists between X and Y, then X and Y are said to be homotopy equivalent or have the same homotopy type, denoted by X≃Y
Example: The unit interval [0,1] and the unit S1 are homotopy equivalent via the maps f(t)=e2πit and g(z)=2πi1logz
Homotopy equivalent spaces share many topological properties, such as the same fundamental group, homology groups, and cohomology rings
Equivalence Relation of Homotopy
Reflexivity
For any topological space X, the identity map idX:X→X is a homotopy equivalence, as idX∘idX=idX≃idX
The constant homotopy H(x,t)=idX(x) demonstrates that idX is homotopic to itself
Symmetry
If f:X→Y is a homotopy equivalence with homotopy inverse g:Y→X, then g is also a homotopy equivalence with homotopy inverse f, as f∘g≃idY and g∘f≃idX
The homotopies demonstrating f∘g≃idY and g∘f≃idX can be reversed to show g∘f≃idX and f∘g≃idY, respectively
Transitivity
If f:X→Y and g:Y→Z are homotopy equivalences with homotopy inverses f′:Y→X and g′:Z→Y, respectively, then g∘f:X→Z is a homotopy equivalence with homotopy inverse f′∘g′:Z→X
The composition of homotopies and the homotopy inverses demonstrate that (g∘f)∘(f′∘g′)≃idZ and (f′∘g′)∘(g∘f)≃idX
Example: If X≃Y and Y≃Z, then X≃Z by composing the homotopy equivalences
Homotopy in Topological Classification
Classifying Spaces by Homotopy Type
Homotopy provides a way to classify topological spaces by grouping together spaces that have the same homotopy type
Spaces that are homotopy equivalent share many important topological invariants, such as the fundamental group, homology groups, and cohomology rings
Example: All contractible spaces (spaces homotopy equivalent to a point) form a single homotopy equivalence class
Homotopy equivalence is a weaker notion than homeomorphism, as homeomorphic spaces are always homotopy equivalent, but homotopy equivalent spaces may not be homeomorphic
Example: The punctured plane R2∖{0} and the unit circle S1 are homotopy equivalent but not homeomorphic
Homotopy Groups and Distinguishing Spaces
The study of homotopy groups, particularly the fundamental group, helps distinguish between spaces that are not homotopy equivalent and provides insight into the structure of topological spaces
The fundamental group π1(X) is a homotopy invariant that captures information about the loops in a space X
Higher homotopy groups πn(X) capture information about higher-dimensional spheres in X
Example: The fundamental group distinguishes between the T2 and the sphere S2, as π1(T2)=Z×Z and π1(S2)=0
Homotopy theory is a powerful tool in algebraic topology, allowing for the development of invariants and techniques to study and classify topological spaces
Key Terms to Review (16)
Circle: A circle is a fundamental geometric shape defined as the set of all points in a plane that are equidistant from a fixed center point. In the context of topology, circles serve as important examples in studying properties like homotopy and fundamental groups, providing a rich framework for understanding continuous deformations and loops within spaces.
Continuous Map: A continuous map is a function between topological spaces that preserves the notion of closeness, meaning that the pre-image of every open set is open. This fundamental concept is crucial as it allows for the comparison of topological spaces and the study of their properties through deformation and transformation. Continuous maps play a central role in defining and understanding homotopy, the fundamental group, and various homological theories.
CW complex: A CW complex is a type of topological space that is built from a set of cells, which are attached together in a specific way. This construction allows for a combinatorial approach to topology, making it easier to study properties like homotopy and homology. CW complexes can be used to analyze more complex spaces by breaking them down into simpler building blocks, facilitating computations in algebraic topology.
Deformation retract: A deformation retract is a type of homotopy equivalence where a topological space can be continuously 'shrunk' to a subspace while keeping the subspace within the space. This means there exists a continuous map from the space to the subspace that behaves like an identity map on the subspace and can be continuously deformed into it. This concept connects to CW complexes and cellular maps as it illustrates how complex spaces can be simplified, while in the context of homotopy, it shows how spaces can be identified based on their topological properties.
Homotopic maps: Homotopic maps are continuous functions between topological spaces that can be continuously deformed into one another, meaning there exists a continuous family of functions that connects them. This concept is fundamental in understanding the idea of homotopy equivalence, which classifies spaces based on their topological properties rather than their precise shape or form. It allows mathematicians to explore the relationships between different spaces and how they can be transformed into one another through continuous transformations.
Homotopy: Homotopy is a concept in algebraic topology that describes a continuous deformation between two continuous functions defined on topological spaces. This idea is foundational for understanding how spaces can be transformed into each other and plays a crucial role in classifying spaces based on their shape and connectivity.
Homotopy Equivalence: Homotopy equivalence is a concept in topology that describes a relationship between two topological spaces where they can be continuously deformed into each other through a series of mappings. This idea helps establish when two spaces share the same topological properties, often used to analyze spaces in relation to their homology and homotopy characteristics.
Homotopy pushout: A homotopy pushout is a construction in algebraic topology that combines spaces along a shared subspace to create a new space that reflects the 'homotopical' nature of the original spaces. This process allows us to understand how maps between spaces behave under continuous deformation, which is essential when studying homotopy equivalences. It provides a way to glue spaces together while preserving their homotopical properties, making it a powerful tool in the analysis of topological spaces.
Invariance of Homotopy Type: Invariance of homotopy type refers to the property that a topological space's homotopy type remains unchanged under homotopy equivalences. This means if two spaces can be continuously transformed into one another, they share the same essential topological features, even if their geometric representations differ. This concept is central to understanding how different spaces can be classified and compared based on their underlying structure rather than their specific shape.
Mapping cone: A mapping cone is a construction in algebraic topology that associates to a continuous map between topological spaces a new space that intuitively represents the 'cylinder' of the map. This concept helps in understanding how spaces change under continuous transformations and plays a crucial role in analyzing homotopy equivalences and fiber sequences, providing a way to study the topological properties of spaces when maps are involved.
Path homotopy: Path homotopy is a concept in algebraic topology that describes when two continuous paths in a topological space can be continuously deformed into one another without leaving the space. This idea is crucial for understanding how paths behave in relation to the topology of the space, as it allows us to classify paths based on their equivalence under deformation. It establishes a notion of 'sameness' among paths, which is essential for discussing homotopy equivalence between spaces.
Quotient Space: A quotient space is a topological space that is formed by taking another space and identifying points together according to an equivalence relation. This process essentially groups points into sets, where each set is represented by a single point in the new space, allowing for a simplification of complex spaces and structures. Quotient spaces are crucial in understanding homotopy equivalence, as they can help reveal the essential shape and structure of a space by collapsing certain features.
Retract: A retract is a continuous map from a topological space to a subspace that is homotopic to the inclusion map of that subspace. This concept is crucial in understanding how spaces can be simplified or reduced while retaining their essential properties. Retracts help in analyzing the relationships between different topological spaces and their structures, especially in relation to deformation retraction and homotopy equivalence.
Simplex: A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, an n-simplex is a convex hull of its n+1 vertices, which can be thought of as the simplest possible polytope in that dimension. This concept connects deeply with the study of topological spaces and forms the building blocks for simplicial complexes, which are used to analyze homotopy and homology.
Suspension: Suspension is a topological operation that takes a space and stretches it into a higher-dimensional object by collapsing its ends into points. This process is pivotal in understanding various concepts in algebraic topology, including how spaces can be transformed and related to each other, providing insights into the structure and properties of CW complexes, homotopy theory, and higher homotopy groups.
Torus: A torus is a doughnut-shaped surface that can be mathematically represented as the Cartesian product of two circles, denoted as $S^1 \times S^1$. This shape is fundamental in algebraic topology, serving as a classic example of a non-trivial topological space that can help illustrate various concepts such as homotopy, homology, and the structure of CW complexes.