1.6 Homotopy invariance

Homotopy invariance is a key concept in cohomology theory. It shows that cohomology groups remain unchanged when spaces are continuously deformed. This property allows us to study complex spaces by examining simpler, homotopy equivalent ones.

Homotopy equivalence is weaker than homeomorphism but still preserves many topological properties. It's crucial for computing cohomology groups of spaces that can be deformed into well-understood shapes, simplifying complex calculations in algebraic topology.

Homotopy and homotopy equivalence

• Homotopy is a fundamental concept in algebraic topology that allows for the continuous deformation of maps and spaces
• Homotopy equivalence is a relation between spaces that preserves their topological properties up to continuous deformation
• Homotopy invariance of cohomology is a crucial property that ensures cohomology groups remain unchanged under homotopy equivalence

Homotopy of maps

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• A homotopy between two continuous maps $f, g: X \to Y$ is a continuous function $H: X \times [0, 1] \to Y$ such that $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x \in X$
• Homotopy of maps captures the idea of continuously deforming one map into another while keeping the endpoints fixed
• If a homotopy exists between two maps, they are said to be homotopic (denoted by $f \simeq g$)
• Homotopy defines an equivalence relation on the set of continuous maps between two spaces

Homotopy equivalence of spaces

• Two topological spaces $X$ and $Y$ are homotopy equivalent (denoted by $X \simeq Y$) if there exist continuous maps $f: X \to Y$ and $g: Y \to X$ such that $g \circ f \simeq id_X$ and $f \circ g \simeq id_Y$
• Homotopy equivalence is a weaker notion than homeomorphism but still preserves many topological properties
• Examples of homotopy equivalent spaces include the circle and the annulus, the real line and the real plane, and the solid torus and the circle

Homotopy invariance of cohomology

• Cohomology theory assigns algebraic objects (abelian groups or rings) to topological spaces in a functorial manner
• Homotopy invariance states that if two spaces $X$ and $Y$ are homotopy equivalent, then their cohomology groups are isomorphic: $H^n(X) \cong H^n(Y)$ for all $n$
• This property allows for the computation of cohomology groups of complicated spaces by studying simpler, homotopy equivalent spaces
• Homotopy invariance is a key feature of cohomology theories such as singular cohomology, de Rham cohomology, and Čech cohomology

Contractible spaces and homotopy

• Contractible spaces are a special class of topological spaces that are homotopy equivalent to a point
• The notion of contractibility is closely related to homotopy and provides a way to identify "trivial" spaces from a homotopy perspective

Contractible spaces

• A topological space $X$ is contractible if the identity map $id_X: X \to X$ is homotopic to a constant map $c: X \to X$, where $c(x) = x_0$ for some fixed point $x_0 \in X$
• Intuitively, a contractible space can be continuously deformed to a single point within itself
• Examples of contractible spaces include the Euclidean space $\mathbb{R}^n$, the unit ball $B^n$, and the cone $CX$ of any topological space $X$
• Contractible spaces have trivial homotopy groups and cohomology groups: $\pi_n(X) = 0$ and $H^n(X) = 0$ for all $n > 0$

Homotopy and contractibility

• If a space $X$ is contractible, then it is homotopy equivalent to a point
• Conversely, if a space $X$ is homotopy equivalent to a point, then it is contractible
• Contractibility is a homotopy invariant property: if $X$ and $Y$ are homotopy equivalent and $X$ is contractible, then $Y$ is also contractible
• The cone construction $CX$ provides a way to create a contractible space from any topological space $X$

Homotopy invariance proofs

• To establish homotopy invariance of cohomology theories, one needs to prove that homotopy equivalent spaces have isomorphic cohomology groups
• The proofs typically involve constructing explicit isomorphisms between the cohomology groups using the homotopy equivalence data

Proof for singular cohomology

• Given a homotopy equivalence $f: X \to Y$ and $g: Y \to X$ with homotopies $H: g \circ f \simeq id_X$ and $K: f \circ g \simeq id_Y$, one constructs chain homotopies between the induced maps on singular cochains
• The chain homotopies induce isomorphisms on the level of cohomology groups $H^n(Y) \to H^n(X)$ and $H^n(X) \to H^n(Y)$, proving homotopy invariance

Proof for de Rham cohomology

• For smooth manifolds $M$ and $N$, a homotopy equivalence induces an isomorphism between their de Rham cohomology groups $H^n_{dR}(M) \cong H^n_{dR}(N)$
• The proof involves constructing a chain homotopy between the pullback maps on differential forms using the homotopy operator
• The chain homotopy induces an isomorphism on the level of de Rham cohomology groups

Proof for Čech cohomology

• Čech cohomology is defined using open covers and their refinements
• To prove homotopy invariance, one shows that a homotopy equivalence induces a morphism between the direct systems of Čech cohomology groups corresponding to open covers
• The induced morphism is an isomorphism on the direct limit, establishing homotopy invariance of Čech cohomology

Applications of homotopy invariance

• Homotopy invariance of cohomology has numerous applications in algebraic topology and related fields
• It allows for the study of topological properties and the computation of invariants using homotopy equivalent spaces

Topological invariants

• Homotopy invariance implies that cohomology groups are topological invariants, meaning they remain unchanged under homeomorphisms
• Other homotopy invariant quantities include homotopy groups, homology groups, and characteristic classes
• These invariants can be used to distinguish non-homeomorphic spaces and study their topological properties

Cohomology of homotopy equivalent spaces

• Homotopy invariance simplifies the computation of cohomology groups for spaces that are homotopy equivalent to well-understood spaces
• For example, the cohomology of the punctured plane $\mathbb{R}^2 \setminus \{0\}$ can be computed using its homotopy equivalence to the circle $S^1$
• Homotopy invariance allows for the development of computational techniques, such as the Mayer-Vietoris sequence, which relate the cohomology of a space to the cohomology of its subspaces

Homotopy invariance vs homeomorphism invariance

• While homotopy invariance and homeomorphism invariance are both important properties in algebraic topology, they differ in their strength and implications

Differences in invariance properties

• Homeomorphism invariance is a stronger property than homotopy invariance
• If two spaces are homeomorphic, they are necessarily homotopy equivalent, but the converse is not true
• Homeomorphism invariant properties include topological properties such as compactness, connectedness, and dimension
• Homotopy invariant properties, such as homotopy groups and cohomology groups, are also homeomorphism invariant

Examples distinguishing homotopy and homeomorphism

• The circle $S^1$ and the annulus $A = \{(x, y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4\}$ are homotopy equivalent but not homeomorphic
• The real line $\mathbb{R}$ and the real plane $\mathbb{R}^2$ are homotopy equivalent but not homeomorphic
• The torus $T^2$ and the wedge sum of circles $S^1 \vee S^1$ are not homotopy equivalent, but they have isomorphic fundamental groups
• These examples demonstrate that homotopy equivalence is a weaker relation than homeomorphism, capturing topological properties up to continuous deformation