🧬Cohomology Theory Unit 7 – Cohomology Operations: Steenrod Squares

Key Concepts and Definitions

• Cohomology operations are natural transformations between cohomology functors that provide additional structure and information beyond the cohomology groups themselves
• Steenrod squares, denoted by $Sq^i$, are cohomology operations that map the $n$-th cohomology group $H^n(X; \mathbb{F}_2)$ to the $(n+i)$-th cohomology group $H^{n+i}(X; \mathbb{F}_2)$ for a topological space $X$ and coefficients in the field $\mathbb{F}_2$
  • The index $i$ represents the degree of the Steenrod square operation
  • Steenrod squares are essential tools in algebraic topology for studying the cohomology of spaces with $\mathbb{F}_2$ coefficients
• The Steenrod algebra, denoted by $\mathcal{A}_2$, is the graded algebra generated by the Steenrod squares subject to certain relations called the Adem relations
  • The Steenrod algebra acts on the cohomology of spaces, providing additional structure and operations beyond the cup product
• The total Steenrod square $Sq = \sum_{i \geq 0} Sq^i$ is the sum of all Steenrod squares and satisfies the Cartan formula, which relates the total Steenrod square of a cup product to the cup product of the total Steenrod squares of the factors
• The excess of a Steenrod square $Sq^i$ is defined as the difference between the degree of the operation and the degree of the cohomology class it acts on, i.e., $e(Sq^i) = i$
• The unstable module over the Steenrod algebra is a graded module equipped with an action of the Steenrod algebra that satisfies the unstability condition, which states that $Sq^i(x) = 0$ if $i > |x|$, where $|x|$ denotes the degree of the element $x$

Historical Context and Development

• The study of cohomology operations originated in the work of Norman Steenrod in the 1940s, who introduced the Steenrod squares as a tool for investigating the cohomology of spaces with coefficients in the field $\mathbb{F}_2$
• Steenrod's motivation for developing cohomology operations came from the desire to understand the relationship between the cohomology of a space and its homotopy type
  • He observed that spaces with isomorphic cohomology groups could have different homotopy types, and cohomology operations provided a way to distinguish them
• The Adem relations, discovered by José Adem in the 1950s, provided a complete set of relations among the Steenrod squares and led to the definition of the Steenrod algebra
• In the 1960s, the concept of cohomology operations was generalized to other coefficient rings, leading to the development of the Steenrod-Epstein operations for odd primes and the Dyer-Lashof operations for the homology of infinite loop spaces
• The work of John Milnor, John Moore, and others in the 1960s and 1970s established the connection between cohomology operations and the structure of the Steenrod algebra, leading to the development of the Adams spectral sequence and other powerful tools in algebraic topology
• The study of cohomology operations has had significant applications in various areas of mathematics, including homotopy theory, characteristic classes, and the classification of manifolds

Algebraic Foundations

• The Steenrod squares are defined using the cohomology of the Eilenberg-MacLane spaces $K(\mathbb{F}_2, n)$, which are spaces with a single non-trivial homotopy group $\pi_n(K(\mathbb{F}_2, n)) \cong \mathbb{F}_2$
  • The cohomology of these spaces, $H^*(K(\mathbb{F}_2, n); \mathbb{F}_2)$, is a polynomial algebra on a single generator $\iota_n$ in degree $n$
• The Steenrod squares are uniquely determined by their action on the generators $\iota_n$ of the cohomology of Eilenberg-MacLane spaces and the requirement that they commute with the cohomology suspension maps
• The Adem relations provide a complete set of relations among the Steenrod squares and are essential for computing the action of Steenrod squares on cohomology classes
  • The Adem relations state that for $a < 2b$, $Sq^a Sq^b = \sum_{c=0}^{\lfloor a/2 \rfloor} \binom{b-c-1}{a-2c} Sq^{a+b-c} Sq^c$
• The Steenrod algebra $\mathcal{A}_2$ is a graded connected Hopf algebra, which means it has a compatible multiplication, comultiplication, and antipode structure
  • The Hopf algebra structure of $\mathcal{A}_2$ is essential for studying the structure of the Steenrod algebra and its dual, the Dyer-Lashof algebra
• The mod 2 cohomology of the classifying space of a group, $H^*(BG; \mathbb{F}_2)$, is a module over the Steenrod algebra, and the action of Steenrod squares on this cohomology provides important information about the group $G$
• The Bockstein homomorphism, which relates the mod 2 cohomology to the integral cohomology, commutes with the Steenrod squares and provides a connection between the action of Steenrod squares and the torsion in the integral cohomology

Construction of Steenrod Squares

• The construction of Steenrod squares relies on the properties of the cohomology of Eilenberg-MacLane spaces and the naturality of cohomology operations
• For each degree $i$, the Steenrod square $Sq^i$ is defined as the unique cohomology operation that satisfies the following properties:
  1. $Sq^i$ is a natural transformation from the functor $H^n(-, \mathbb{F}_2)$ to the functor $H^{n+i}(-, \mathbb{F}_2)$
  2. $Sq^i(\iota_n) = \iota_{n+i}$ for the generator $\iota_n \in H^n(K(\mathbb{F}_2, n); \mathbb{F}_2)$
  3. $Sq^i$ commutes with the cohomology suspension maps
• The existence and uniqueness of Steenrod squares satisfying these properties are guaranteed by the representability of cohomology functors and the Yoneda lemma
• The Cartan formula, which relates the Steenrod squares of a cup product to the cup product of Steenrod squares, is a consequence of the naturality of Steenrod squares and their action on the cohomology of Eilenberg-MacLane spaces
  • The Cartan formula states that for cohomology classes $x$ and $y$, $Sq^k(x \cup y) = \sum_{i+j=k} Sq^i(x) \cup Sq^j(y)$
• The Adem relations among Steenrod squares are derived using the cohomology of the Eilenberg-MacLane spaces and the naturality of cohomology operations
• The construction of Steenrod squares can be generalized to other cohomology theories, such as $K$-theory and cobordism theory, by replacing the Eilenberg-MacLane spaces with the appropriate classifying spaces

Properties and Axioms

• The Steenrod squares satisfy a set of axioms that characterize their behavior and properties:
  1. Naturality: Steenrod squares commute with cohomology homomorphisms induced by continuous maps between spaces
  2. Stability: Steenrod squares commute with the cohomology suspension maps
  3. Normalization: $Sq^0$ is the identity homomorphism, and $Sq^i(\iota_n) = \iota_{n+i}$ for the generator $\iota_n \in H^n(K(\mathbb{F}_2, n); \mathbb{F}_2)$
  4. Cartan formula: For cohomology classes $x$ and $y$, $Sq^k(x \cup y) = \sum_{i+j=k} Sq^i(x) \cup Sq^j(y)$
  5. Adem relations: For $a < 2b$, $Sq^a Sq^b = \sum_{c=0}^{\lfloor a/2 \rfloor} \binom{b-c-1}{a-2c} Sq^{a+b-c} Sq^c$
• The Steenrod squares are compatible with the cup product structure in cohomology, as evident from the Cartan formula
  • This compatibility makes Steenrod squares a powerful tool for studying the multiplicative structure of cohomology rings
• The Adem relations provide a complete set of relations among the Steenrod squares and are essential for computing the action of Steenrod squares on cohomology classes
• The excess of a Steenrod square, defined as the difference between the degree of the operation and the degree of the cohomology class it acts on, plays a crucial role in the unstability condition for modules over the Steenrod algebra
  • The unstability condition states that $Sq^i(x) = 0$ if $i > |x|$, where $|x|$ denotes the degree of the element $x$
• The Steenrod squares satisfy the Bockstein relation, which relates the Bockstein homomorphism $\beta$ to the Steenrod squares: $\beta Sq^i = Sq^i \beta + Sq^{i-1}$
  • This relation connects the action of Steenrod squares to the torsion in the integral cohomology
• The Steenrod squares are derivations with respect to the cup product, meaning that they satisfy the Leibniz rule: $Sq^k(x \cup y) = Sq^k(x) \cup y + x \cup Sq^k(y)$

Applications in Topology

• Steenrod squares are essential tools in algebraic topology for studying the cohomology of spaces and their relationships to homotopy theory
• The action of Steenrod squares on the cohomology of a space provides additional information about its homotopy type beyond the cohomology groups themselves
  • Spaces with isomorphic cohomology groups but different Steenrod square actions have different homotopy types
• Steenrod squares are used to define and compute characteristic classes of vector bundles, such as the Stiefel-Whitney classes and the Wu classes
  • These characteristic classes provide important invariants for studying the topology of manifolds and their embeddings
• The Steenrod squares play a crucial role in the Adams spectral sequence, which is a powerful tool for computing stable homotopy groups of spheres and other spaces
  • The $E_2$-term of the Adams spectral sequence is the cohomology of the Steenrod algebra, and the differentials are determined by the action of Steenrod squares
• Steenrod squares are used in the study of the cohomology of classifying spaces of groups, providing information about the structure and properties of the group
  • The action of Steenrod squares on the cohomology of the classifying space $H^*(BG; \mathbb{F}_2)$ is related to the group cohomology and the Quillen cohomology of the group $G$
• The Steenrod squares are essential in the proof of the Hopf invariant one problem, which states that the only spheres that admit a division algebra structure are $S^0$, $S^1$, $S^3$, and $S^7$
  • The proof relies on the analysis of the action of Steenrod squares on the cohomology of the spheres and their loop spaces
• Steenrod squares have applications in the study of the topology of manifolds, such as the intersection forms of 4-manifolds and the Kervaire invariant problem

Computational Techniques

• Computing the action of Steenrod squares on cohomology classes is a fundamental problem in algebraic topology, and several computational techniques have been developed to address this problem
• The Adem relations provide a basis for computing the action of Steenrod squares on cohomology classes by expressing higher-degree squares in terms of lower-degree ones
  • To compute $Sq^k(x)$ for a cohomology class $x$, one can use the Adem relations to express $Sq^k$ as a sum of compositions of lower-degree squares and then apply them to $x$
• The Cartan formula is another essential tool for computing Steenrod squares of cup products, as it reduces the computation to the Steenrod squares of the individual factors
• The Wu formula expresses the Stiefel-Whitney classes of a manifold in terms of the Steenrod squares of the fundamental class, providing a method for computing these characteristic classes
  • The Wu formula states that $w_i = \sum_{j=0}^i Sq^j(v_{i-j})$, where $w_i$ is the $i$-th Stiefel-Whitney class and $v_i$ is the $i$-th Wu class
• The Bullett-Macdonald identity is a useful formula for computing the Steenrod squares of the mod 2 reduction of integral cohomology classes
  • The identity relates the Steenrod squares of the mod 2 reduction to the Steenrod squares of the integral class and the Bockstein homomorphism
• Spectral sequences, such as the Serre spectral sequence and the Adams spectral sequence, are powerful tools for computing cohomology groups and the action of Steenrod squares
  • These spectral sequences often involve the cohomology of the Steenrod algebra and the action of Steenrod squares on the $E_2$-term
• Computer algebra systems, such as Sage and Magma, have implemented algorithms for computing Steenrod squares and their actions on cohomology classes
  • These systems provide efficient methods for performing computations in the Steenrod algebra and its modules

Advanced Topics and Extensions

• The study of Steenrod squares and cohomology operations has led to the development of several advanced topics and extensions in algebraic topology
• The Adams spectral sequence is a generalization of the Serre spectral sequence that uses the Steenrod algebra and its cohomology to compute stable homotopy groups of spaces
  • The $E_2$-term of the Adams spectral sequence is the cohomology of the Steenrod algebra, and the differentials are determined by the action of Steenrod squares
• The Dyer-Lashof operations are cohomology operations for the homology of infinite loop spaces, analogous to the Steenrod squares for the cohomology of spaces
  • The Dyer-Lashof operations are essential for studying the structure of the homology of iterated loop spaces and the structure of the dual Steenrod algebra
• The Steenrod-Epstein operations are cohomology operations for odd primes, analogous to the Steenrod squares for the prime 2
  • These operations are defined using the cohomology of the Eilenberg-MacLane spaces with coefficients in the field $\mathbb{F}_p$ for odd primes $p$
• The Milnor primitives are a family of cohomology operations that generate the dual Steenrod algebra and provide a basis for studying the structure of the Steenrod algebra
  • The Milnor primitives are related to the Dickson invariants and the Dyer-Lashof operations
• The study of the Steenrod algebra and its modules has led to the development of the theory of unstable modules and the Lannes-Zarati functor
  • The Lannes-Zarati functor is a powerful tool for studying the structure of unstable modules over the Steenrod algebra and their relationships to the cohomology of spaces
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