5.3 The Hurewicz theorem

The Hurewicz theorem connects homotopy and homology groups, revealing deep relationships between these fundamental concepts in algebraic topology. It shows that for simply connected spaces, the first non-trivial homotopy group matches the first non-trivial homology group.

This powerful result simplifies the computation of homotopy groups, which are often harder to calculate directly. By leveraging more accessible homology calculations, the Hurewicz theorem provides a crucial tool for understanding the topological structure of spaces.

Hurewicz Theorem: Homotopy vs Homology

Relationship between Homotopy and Homology Groups

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• The Hurewicz theorem establishes a relationship between the homotopy groups and homology groups of a space, under certain connectedness conditions
• For a simply connected space X, the Hurewicz theorem states that the first non-trivial homotopy group is isomorphic to the first non-trivial homology group
  • Mathematically, this is expressed as $\pi_n(X) \cong H_n(X)$ for the smallest $n \geq 2$ such that $\pi_n(X)$ is non-trivial
  • Example: If X is a simply connected space with $\pi_2(X) \cong \mathbb{Z}$ and $\pi_n(X) = 0$ for $n > 2$, then $H_2(X) \cong \mathbb{Z}$ and $H_n(X) = 0$ for $n > 2$

Proof of the Hurewicz Theorem

• The proof of the Hurewicz theorem involves the construction of the Hurewicz homomorphism, which maps elements of the homotopy group to elements of the homology group
  • The Hurewicz homomorphism is defined using the concept of singular homology, where a continuous map from the n-simplex to the space X is considered
  • The proof shows that the Hurewicz homomorphism is an isomorphism under the given conditions, using techniques from algebraic topology
• Key steps in the proof include:
  • Constructing the Hurewicz homomorphism $h_n: \pi_n(X) \to H_n(X)$
  • Showing that $h_n$ is a group homomorphism
  • Proving that $h_n$ is an isomorphism using the long exact sequence of a pair and the relative Hurewicz theorem
  • Applying the connectivity conditions to ensure the isomorphism holds for the first non-trivial homotopy and homology groups

Computing Homotopy Groups with Hurewicz

Determining Connectivity and Homology Groups

• To compute the first non-trivial homotopy group of a space X using the Hurewicz theorem, one needs to determine the connectivity of the space and find the first non-trivial homology group
• If X is simply connected (i.e., $\pi_1(X) = 0$), then the Hurewicz theorem applies
  • Example: The 2-dimensional sphere $S^2$ is simply connected, so the Hurewicz theorem can be used to compute its first non-trivial homotopy group
• The homology groups of a space can be computed using various techniques, such as cellular homology or singular homology, depending on the properties of the space
  • Example: For a CW complex, cellular homology can be used to calculate the homology groups efficiently

Applying the Hurewicz Theorem

• Once the first non-trivial homology group $H_n(X)$ is determined, the Hurewicz theorem implies that $\pi_n(X) \cong H_n(X)$, thus providing the first non-trivial homotopy group
• If the homology groups are known, the Hurewicz theorem allows for the immediate computation of the corresponding homotopy group
  • Example: If $H_3(X) \cong \mathbb{Z}_2$ is the first non-trivial homology group of a simply connected space X, then $\pi_3(X) \cong \mathbb{Z}_2$ by the Hurewicz theorem
• The Hurewicz theorem simplifies the computation of homotopy groups by reducing the problem to calculating homology groups, which are often more tractable

Hurewicz Theorem for Simply Connected Spaces

Isomorphism between Homotopy and Homology Groups

• For a simply connected space X, the theorem establishes an isomorphism between the first non-trivial homotopy group and the first non-trivial homology group, providing a link between the two fundamental invariants
• This isomorphism allows for the computation of homotopy groups using homology groups, which are often easier to calculate
  • Example: For a simply connected CW complex X, the first non-trivial homotopy group can be determined by calculating the first non-trivial homology group using cellular homology

Implications for the Study of Simply Connected Spaces

• The Hurewicz theorem has significant implications for the study of simply connected spaces in algebraic topology
• The theorem implies that for a simply connected space, the first non-trivial homotopy group determines the first non-trivial homology group, and vice versa
  • Example: If $\pi_4(X) \cong \mathbb{Z}$ for a simply connected space X, then $H_4(X) \cong \mathbb{Z}$, and all lower homotopy and homology groups are trivial
• The Hurewicz theorem provides a foundation for the study of higher-dimensional homotopy groups and their relationship to homology groups in simply connected spaces
  • It allows for the extension of results and techniques from homology theory to the study of homotopy groups
  • The theorem is a crucial tool in the computation and classification of homotopy groups of simply connected spaces

Relative Homotopy Groups with Hurewicz

Relative Hurewicz Theorem

• The relative Hurewicz theorem is an extension of the Hurewicz theorem that relates relative homotopy groups to relative homology groups
• For a pair of spaces $(X, A)$, where A is a subspace of X, the relative Hurewicz theorem states:
  • If X and A are connected and X is obtained from A by attaching cells of dimension $\geq n$, then the relative homotopy group $\pi_n(X, A)$ is isomorphic to the relative homology group $H_n(X, A)$ for $n \geq 2$
  • Mathematically, this is expressed as $\pi_n(X, A) \cong H_n(X, A)$ under the given conditions

Applying the Relative Hurewicz Theorem

• To apply the relative Hurewicz theorem, one needs to identify the connectivity of the spaces X and A, and determine the dimension of the cells attached to A to form X
• The relative homology groups $H_n(X, A)$ can be computed using techniques such as the long exact sequence of a pair or the excision theorem
  • Example: If $(X, A)$ is a CW pair, the relative homology groups can be computed using the cellular chain complex of the pair
• Once the relative homology group $H_n(X, A)$ is determined, the relative Hurewicz theorem implies that $\pi_n(X, A) \cong H_n(X, A)$, thus providing the relative homotopy group
  • Example: If $H_3(X, A) \cong \mathbb{Z}$ and the conditions of the relative Hurewicz theorem are satisfied, then $\pi_3(X, A) \cong \mathbb{Z}$
• The relative Hurewicz theorem allows for the computation of relative homotopy groups by reducing the problem to calculating relative homology groups, which can be more accessible using algebraic techniques