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Steenrod algebra

In algebraic topology, the Steenrod algebra \mathcal{A}_p, for a prime p, is the graded associative algebra over the field \mathbb{F}_p that encodes all natural stable cohomology operations on the mod-p cohomology groups H^*(X; \mathbb{F}_p) of topological spaces X. These operations commute with induced maps from continuous functions and are stable under suspension, meaning they preserve degree shifts in the stable homotopy category. As a Hopf algebra, \mathcal{A}_p admits a coproduct \psi: \mathcal{A}_p \to \mathcal{A}_p \otimes \mathcal{A}_p derived from the diagonal map on spaces, enabling it to act compatibly on cup products in cohomology rings.^[1] For p = 2, \mathcal{A}_2 is generated by the Steenrod squares Sq^i: H^n(X; \mathbb{F}_2) \to H^{n+i}(X; \mathbb{F}_2) for i \geq 0, with Sq^0 the identity, subject to the Adem relations Sq^a Sq^b = \sum_{j \geq 0} \binom{b-j-1}{a-2j} Sq^{a+b-j} Sq^j when $0 < a < 2b. These relations ensure uniqueness, and a basis is given by admissible monomials Sq^{i_1} \cdots Sq^{i_k} where i_r \geq 2 i_{r+1} for each r.^[1] For odd primes p, \mathcal{A}_p is generated by the reduced p-th power operations P^i: H^n(X; \mathbb{F}_p) \to H^{n + i(p-1)}(X; \mathbb{F}_p) for i \geq 0 and the Bockstein operation \beta: H^n(X; \mathbb{F}_p) \to H^{n+1}(X; \mathbb{F}_p), again modulo Adem relations of the form P^a P^b = \sum_{j=0}^{\lfloor a/p \rfloor} (-1)^{a+j} \binom{(p-1)(b-j)-1}{a - p j} P^{a+b-j} P^j for $0 < a < p b, along with \beta^2 = 0 and mixed relations.^[1] The dual algebra \mathcal{A}_p^* is commutative and has a simpler structure as the tensor product of an exterior algebra on generators \tau_i of degree $2p^i - 1 and a polynomial algebra on generators \xi_i of degree $2p^i - 2.^[2] The Steenrod squares were first defined by Norman E. Steenrod in 1947 using explicit cochain formulas for higher cup-products (cup-i products) in mod-2 cohomology of simplicial complexes.^[3] The general framework of the Steenrod algebra emerged in the early 1950s through work by Steenrod, Henri Cartan, and others, with Cartan formalizing the algebra of stable operations in 1955 and proving key iteration properties.^[4] John Milnor's 1958 analysis provided the explicit basis and Hopf algebra structure, resolving longstanding questions about its generators and relations.^[2] These developments, compiled in Steenrod and Epstein's 1962 monograph, established \mathcal{A}_p as a foundational tool.^[1]

Background in Cohomology

Ordinary Mod p Cohomology Rings

Singular cohomology with coefficients in \mathbb{F}_p, the finite field with p elements where p is prime, assigns to each topological space X a sequence of abelian groups H^n(X; \mathbb{F}_p) for n \geq 0, derived as the cohomology of the singular cochain complex C^*(X; \mathbb{F}_p).^[5] This complex consists of \mathbb{F}_p-linear maps from the singular simplicial chains of X to \mathbb{F}_p, with the coboundary operator induced by the boundary maps on chains.^[5] The groups H^*(X; \mathbb{F}_p) form a graded-commutative ring under the cup product, a bilinear operation \cup: H^m(X; \mathbb{F}_p) \times H^n(X; \mathbb{F}_p) \to H^{m+n}(X; \mathbb{F}_p) that is associative and graded-commutative, meaning \alpha \cup \beta = (-1)^{mn} \beta \cup \alpha for \alpha \in H^m, \beta \in H^n.^[5] As \mathbb{F}_p-vector spaces, these cohomology groups are contravariant functors from the category of topological spaces and continuous maps to the category of graded vector spaces, with naturality ensuring that for any continuous map f: X \to Y, the induced homomorphism f^*: H^*(Y; \mathbb{F}_p) \to H^*(X; \mathbb{F}_p) preserves the ring structure.^[5] Additionally, for any short exact sequence of abelian groups serving as coefficients $0 \to A \to B \to C \to 0, there arises a long exact sequence in cohomology \cdots \to H^n(X; A) \to H^n(X; B) \to H^n(X; C) \to H^{n+1}(X; A) \to \cdots.^[5] A representative example is the cohomology of the n-sphere S^n, where H^k(S^n; \mathbb{F}_p) \cong \mathbb{F}_p for k = 0 and k = n, and H^k(S^n; \mathbb{F}_p) = 0 otherwise; the cup product structure is trivial in positive degrees since there is at most one nonzero class up to scalar multiple.^[6] The foundations of ordinary mod p cohomology rings were laid in the 1930s and 1940s, with Witold Hurewicz introducing key concepts in homology for manifolds and spheres around 1935–1936, and Norman Steenrod advancing the theory through cohomology products in the early 1940s, culminating in axiomatic formulations.^[7]

Stable Cohomology Operations

A cohomology operation is a natural transformation between cohomology functors, specifically a family of maps \theta_n: H^n(X; \mathbb{F}_p) \to H^{n+k}(X; \mathbb{F}_p) for all spaces X and integers n, that commutes with the homomorphisms induced by continuous maps f: X \to Y, i.e., \theta_n(f^* \alpha) = f^* \theta_n(\alpha) for \alpha \in H^n(X; \mathbb{F}_p).^[1] These operations enrich the algebraic structure of cohomology groups beyond their ring properties, providing tools to distinguish topological features.^[8] Cohomology operations are classified as unstable or stable based on their behavior under suspension. Unstable operations, such as certain higher-degree maps, do not necessarily commute with the suspension isomorphism \Sigma^*: H^n(X; \mathbb{F}_p) \to H^{n+1}(\Sigma X; \mathbb{F}_p), often vanishing when applied to classes whose degree is less than the operation's degree, e.g., \theta(x) = 0 if \deg \theta > \deg x.^[1] Stable operations, in contrast, satisfy \Sigma^* \theta_n = \theta_{n+1} \Sigma^*, preserving their action across suspensions and thus remaining well-defined in the stable homotopy category.^[1] This stability is crucial because many topological problems, particularly those involving high-dimensional spaces or infinite suspensions, require invariants that are insensitive to dimensional shifts.^[8] Prominent examples of stable operations include the Bockstein operation \beta, which for p=2 coincides with Sq^1 and has degree 1, arising from the connecting homomorphism in the long exact sequence of a fibration or short exact sequence of coefficients. For odd primes p, stable operations include the reduced powers P^k of degree $2(p-1)k, generalizing the action of Frobenius maps on cohomology, as well as the Bockstein \beta of degree 1.^[1] Not all cohomology operations are stable; for instance, higher unstable Steenrod powers fail the suspension commutativity due to dimensional constraints, necessitating a dedicated algebra generated solely by stable ones to systematically detect non-trivial homotopy classes.^[8] The importance of stable cohomology operations lies in their ability to generate an algebra that acts on H^*(X; \mathbb{F}_p) for any space X, enabling the detection of homotopy classes that the cohomology ring structure alone cannot distinguish, such as essential elements in the stable homotopy groups of spheres like the Hopf maps.^[1] By providing stable invariants, these operations facilitate computations in stable homotopy theory, where suspension leads to isomorphisms, and address the limitations of unstable operations that introduce indeterminacy outside specific dimensional ranges.^[8] This algebra of stable operations thus forms the foundation for analyzing complex topological phenomena beyond basic cohomology.^[1]

Definition and Axioms

Axiomatic Characterization

The Steenrod algebra \mathcal{A}_p over the prime field \mathbb{F}_p admits an axiomatic characterization as the unique graded algebra of stable cohomology operations on mod p cohomology rings of topological spaces satisfying a specific set of properties. This characterization captures the essential structure without reference to explicit constructions, emphasizing naturality, multiplicativity, and instability. For p=2, the algebra is generated by the Steenrod squares \mathrm{Sq}^i of degree i \geq 0, while for odd primes p, it is generated by the reduced powers P^i of degree $2i(p-1) and the Bockstein operation \beta of degree 1. These generators satisfy the following axioms, which ensure compatibility with the ring structure and suspension in cohomology.^[1] The axioms for p=2 are: (1) Each \mathrm{Sq}^i: H^n(X;\mathbb{F}_2) \to H^{n+i}(X;\mathbb{F}_2) is a natural transformation of functors from the category of topological spaces and continuous maps to graded \mathbb{F}_2-vector spaces; (2) \mathrm{Sq}^0 is the identity map; (3) For a cohomology class x \in H^n(X;\mathbb{F}_2), \mathrm{Sq}^n(x) = x^2; (4) \mathrm{Sq}^i(x) = 0 if i > n, encoding the instability condition; (5) The Cartan formula holds: \mathrm{Sq}^i(xy) = \sum_{j=0}^i \mathrm{Sq}^j(x) \cdot \mathrm{Sq}^{i-j}(y). Additionally, \mathrm{Sq}^1 corresponds to the connecting homomorphism (Bockstein) associated to the short exact sequence $0 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 0. These properties ensure the operations are stable under suspension, commuting with the suspension isomorphism in cohomology.^[1] For odd p, the axioms are analogous but adjusted for the degrees: (1) Each P^i: H^n(X;\mathbb{F}_p) \to H^{n+2i(p-1)}(X;\mathbb{F}_p) is a natural transformation of functors from the category of topological spaces and continuous maps to graded \mathbb{F}_p-vector spaces; (2) P^0 is the identity; (3) For x \in H^{2k}(X;\mathbb{F}_p), P^k(x) = x^p; (4) P^i(x) = 0 if $2i > n; (5) The Cartan formula: P^i(xy) = \sum_{j=0}^i P^j(x) \cdot P^{i-j}(y). The Bockstein \beta: H^n(X;\mathbb{F}_p) \to H^{n+1}(X;\mathbb{F}_p) satisfies \beta^2 = 0, \beta P^i = P^i \beta (up to sign), and the Leibniz rule \beta(xy) = \beta(x)y + (-1)^{\deg x} x \beta(y), arising from the sequence $0 \to \mathbb{Z}_p \to \mathbb{Z}_{p^2} \to \mathbb{Z}_p \to 0. Stability follows from commutation with suspension.^[1]^[9] The universal property of \mathcal{A}_p asserts that it is the initial object in the category of graded \mathbb{F}_p-algebras equipped with a map to the endomorphism algebra of mod p cohomology rings of spaces, such that the induced operations satisfy the above axioms. Any other algebra of stable operations obeying these properties admits a unique homomorphism to \mathcal{A}_p. Existence and uniqueness of \mathcal{A}_p follow from a functional construction on Eilenberg-MacLane spaces, which is detailed elsewhere.^[9]^[1] These axioms generalize naturally to the context of generalized cohomology theories, particularly the Eilenberg-MacLane spectrum H\mathbb{Z}/p, where \mathcal{A}_p acts as the endomorphism ring of the spectrum, preserving the same relational properties. This spectral perspective unifies the axiomatization across stable homotopy categories.^[9] The axiomatic framework was developed in the 1950s through the work of Henri Cartan and Samuel Eilenberg, building on Steenrod's introduction of squaring operations in ordinary cohomology.^[1]

Generating Operations

The Steenrod algebra at the prime p=2 is generated by the Steenrod squares \mathrm{Sq}^i for i \geq 0, which are stable cohomology operations \mathrm{Sq}^i: H^n(X; \mathbb{F}_2) \to H^{n+i}(X; \mathbb{F}_2).^[10] These operations satisfy \mathrm{Sq}^0 = [\mathrm{id}](/page/id), \mathrm{Sq}^i(x) = x^2 when i = \deg x, and \mathrm{Sq}^i(x) = 0 when i > \deg x, reflecting the instability condition that prevents operations from producing nonzero results beyond the degree of the input class.^[10] On the polynomial algebra \mathbb{F}_2 modeling the cohomology of \mathbb{R}P^\infty, where \deg u = 1, the action of \mathrm{Sq}^i on the monomial u^m is given by \mathrm{Sq}^i(u^m) = \binom{m}{i} u^{m+i} \pmod{2}, with the binomial coefficient computed modulo 2 via Lucas' theorem, which determines when the operation is nonzero based on the binary digits of m and i.^[11] Due to instability, \mathrm{Sq}^i \circ \mathrm{Sq}^j = \mathrm{Sq}^{i+j} when acting on classes x with \deg x < i+j, as both sides vanish.^[10] For odd primes p, the Steenrod algebra is generated by the reduced powers P^i for i \geq 0 and the Bockstein operation \beta, where P^i: H^n(X; \mathbb{F}_p) \to H^{n + 2i(p-1)}(X; \mathbb{F}_p) and \beta: H^n(X; \mathbb{F}_p) \to H^{n+1}(X; \mathbb{F}_p).^[12] The reduced powers satisfy P^0 = \mathrm{id} and P^i(x) = x^p when \deg x = 2i, while \beta arises as the connecting homomorphism in the short exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 0.[12] Instability ensures \beta^\varepsilon P^i(x) = 0for\varepsilon \in {0,1}when\deg x < 2i + \varepsilon$.^[12]

Relations and Identities

Adem Relations

The Adem relations provide the fundamental quadratic relations in the Steenrod algebra that govern the composition of its generating operations, ensuring that the algebra is finite-dimensional in each bidegree. For the mod 2 Steenrod algebra generated by the Steenrod squares \mathrm{Sq}^i with i > 0, the relations state that if $0 < a < 2b, then

\mathrm{Sq}^a \mathrm{Sq}^b = \sum_{c=0}^{\lfloor (2b - a - 1)/2 \rfloor} \binom{b - c - 1}{a - 2c} \mathrm{Sq}^{a + b - c} \mathrm{Sq}^c,

where the binomial coefficients are reduced modulo 2. These relations allow non-admissible compositions to be expressed as linear combinations of admissible ones, where an admissible monomial is of the form \mathrm{Sq}^{i_1} \mathrm{Sq}^{i_2} \cdots \mathrm{Sq}^{i_r} with i_k \geq 2 i_{k+1} for each k, and such monomials (together with the identity \mathrm{Sq}^0) form an additive basis for the algebra.^[13] For an odd prime p, the Steenrod algebra is generated by the power operations P^i with i > 0 (of bidegree (2i(p-1), i)) and the Bockstein operation \beta (of bidegree (1, 0)). The Adem relations include those for compositions of powers: if $0 < a < p b, then

P^a P^b = \sum_{j=0}^{\lfloor a/p \rfloor} (-1)^{a + j} \binom{(p-1)(b - j) - 1}{a - p j} P^{a + b - j} \beta^j \pmod{p},

along with analogous relations for P^a \beta and \beta P^a, which involve multinomial coefficients modulo p to account for the Leibniz rule interactions. Admissible monomials in these generators satisfy i_k \geq p i_{k+1} (ignoring \beta's), and the relations enable any monomial to be rewritten uniquely as a linear combination of admissible basis elements.^[13]^[2] These relations were originally derived by computing the induced operations on the cohomology of products of Eilenberg-MacLane spaces, leveraging the Künneth theorem and properties of cohomology rings; alternatively, they arise from the functional construction of the Steenrod algebra via power series on spectra or from stability in cobordism theories. The relations imply the uniqueness of the axiomatic characterization and confirm the finite dimensionality in each degree.^[13]

Bullett-Macdonald Identities

The Bullett-Macdonald identities offer a generating function reformulation of the Adem relations specifically for the reduced p-th power operations P^i in the mod p Steenrod algebra at odd primes p, enabling the expression of compositions of higher powers in terms of products of lower-degree operations. These identities are unique to odd primes, as the prime-2 case relies on distinct relations for Steenrod squares. The core identity states that the total power operation P(f) = \sum_{i \geq 0} P^i t^i, acting on cohomology classes, satisfies

P(s) \circ P(1) = P(u) \circ P(t^p),

where s = t u and u = \sum_{i=0}^{p-1} t^i = (1 - t^p)/(1 - t). This equation holds as natural transformations between cohomology functors and arises from the multiplicativity and additivity axioms of the Steenrod operations in the algebraic construction of the algebra as endomorphisms of the functor V \mapsto \mathbb{F}_p[V].^[14] Expanding both sides in powers of t yields explicit relations for compositions P^i P^j. For instance, when i < p j, the left side contributes terms involving P^{i+j} + lower compositions, while the right side expresses them via admissible forms P^k P^l with k \geq p l, with coefficients determined by residues of rational functions or binomial expansions mod p. This reformulation simplifies derivations compared to direct binomial coefficient computations in the Adem relations, as the generating form avoids memorizing intricate combinatorial factors. The identities can also be derived via change-of-rings isomorphisms in the cobar complex resolution of the Eilenberg-MacLane spectrum, confirming their consistency with the Hopf algebra structure.^[14] A significant application concerns the action on p-th powers of classes. For an odd prime p and a cohomology class x with P^{a_1 + \cdots + a_k}(x^{p^m}), where each a_i < p, the Bullett-Macdonald identities imply a decomposition using the base-p digits of the exponents: P^{a_1 + \cdots + a_k}(x^{p^m}) = \sum P^{b_1} \cdots P^{b_k}(x)^{p^{m'}} , where the b_i are the base-p digits of the total exponent adjusted for the power level m' \leq m, and the sum runs over distributions compatible with the excess (the internal degree shift). This follows from iteratively applying the commutation property P^i(y^p) = [P^i(y)]^p combined with the generating identity to "peel off" lower powers. For example, at p=3 and m=1, P^2(x^3) = P^2(x)^3 + \binom{3}{2}_3 P^1(x)^3 x^{2(p-1)} + \cdots, but the full expansion reduces via the identity to terms involving only P^0 and P^1 raised to powers, with coefficients vanishing unless digit conditions hold. Such decompositions highlight how higher operations on iterated p-th powers reduce to products of primitive P^i (i < p) applied to x and then raised to p-th powers.^[14]^[15] These identities connect to the Dyer-Lashof algebra, the universal quotient of the free algebra on excess-preserving operations, where the Steenrod algebra embeds as a sub-Hopf algebra; the Bullett-Macdonald relations enforce the necessary kernel for this quotient, ensuring compatibility with excess bounds in homology operations. Overall, they underscore that the Steenrod algebra at odd p is generated by \{P^i \mid i > 0\} and the Bockstein \beta subject to the Adem relations (equivalently, these identities) and the relations involving compositions with the Bockstein \beta.

Constructions

Functional Construction

The functional construction of the Steenrod algebra \mathcal{A}_p realizes it as the graded algebra of stable homotopy classes of self-maps of the Eilenberg-MacLane spectrum H\mathbb{F}_p, where \mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}. Specifically, \mathcal{A}_p = [H\mathbb{F}_p, H\mathbb{F}_p]_*, the stable homotopy groups of the mapping spectrum \mathrm{Map}(H\mathbb{F}_p, H\mathbb{F}_p), graded by the degree of the maps. This identifies \mathcal{A}_p with the endomorphism ring \pi_*(F(H\mathbb{F}_p, H\mathbb{F}_p)) in the stable homotopy category, where elements of degree k correspond to [H\mathbb{F}_p, \Sigma^k H\mathbb{F}_p].^[8]^[16] The generators of \mathcal{A}_p arise naturally from this construction. For p=2, the Steenrod squares \mathrm{Sq}^i (for i \geq 0) are represented by the stable classes [\mathrm{id}, \Sigma^i] in [H\mathbb{F}_2, \Sigma^i H\mathbb{F}_2], which on the underlying Eilenberg-MacLane spaces K(\mathbb{F}_2, n) induce the cup-i square operations via attaching cells or multiplication in cohomology. For odd primes p, the Bockstein \beta and power operations P^i are similarly defined, with \beta: H\mathbb{F}_p \to \Sigma H\mathbb{F}_p and P^i: H\mathbb{F}_p \to \Sigma^{i(p-1)} H\mathbb{F}_p. These generators induce the corresponding Steenrod operations on mod-p cohomology. They satisfy the axioms of the Steenrod algebra, as characterized in the axiomatic approach.^[8]^[17]^[16] The axioms follow directly from the topological structure. Naturality holds because self-maps of H\mathbb{F}_p induce natural transformations on the cohomology functor H^*(-; \mathbb{F}_p), as the mapping space construction is functorial in the domain spectrum. The Cartan formula, \theta^k(xy) = \sum_{i+j=k} \theta^i(x) \cup \theta^j(y) for \theta^k \in \{\mathrm{Sq}^k, P^k, \beta\}, arises from the smash product decomposition: the cup product in cohomology corresponds to the smash product of spectra, H\mathbb{F}_p \wedge H\mathbb{F}_p \simeq H\mathbb{F}_p, with the diagonal coaction induced by the ring spectrum structure.^[8]^[16] Geometrically, elements of \mathcal{A}_p act as stable maps \Sigma^\infty X_+ \to \Sigma^\infty X_+ over H\mathbb{F}_p for simply connected spaces X, where the map factors through the unit \Sigma^\infty X_+ \to H\mathbb{F}_p and induces the cohomology operation on [H\mathbb{F}_p, \Sigma^\infty X_+]_* \cong H^*(X; \mathbb{F}_p). This interprets operations as equivariant maps in the category of H\mathbb{F}_p-modules. The dual Steenrod algebra \mathcal{A}_p^\vee emerges via the co-H-space structure on H\mathbb{F}_p, where the coproduct \psi: \mathcal{A}_p \to \mathcal{A}_p \otimes \mathcal{A}_p dualizes to the coaction on homology H_*(X; \mathbb{F}_p), yielding \mathcal{A}_p^\vee \cong \mathbb{F}_p[\xi_i] \otimes \Lambda_{\mathbb{F}_p}(\tau_j) (mod p) or \mathbb{F}_2[\zeta_i] (mod 2).^[8]^[17]^[16]

Algebraic Construction

The mod 2 Steenrod algebra \mathcal{A}_2 admits an algebraic presentation as the quotient of the tensor algebra T(V) over \mathbb{F}_2, where V is the graded vector space with basis \{\sigma_n \mid n \geq 1\} and \deg(\sigma_n) = n, by the two-sided ideal generated by the Adem relations. This presentation realizes \mathcal{A}_2 as a graded associative algebra generated by the symbols \sigma_n, corresponding to the Steenrod squares \mathrm{Sq}^n, with the relations ensuring consistency with the stable cohomology operations. For a general prime p, an explicit algebraic construction of the Steenrod algebra \mathcal{A}_p is obtained as the graded dual Hopf algebra of the commutative Hopf algebra \overline{\mathcal{A}}_p described by Milnor.^[18] Specifically, \overline{\mathcal{A}}_p is the tensor product of an exterior algebra E(\tau_0, \tau_1, \dots) on generators \tau_i (i \geq 0) of degree $2p^i - 1 and a polynomial algebra P(\xi_1, \xi_2, \dots) on generators \xi_i (i \geq 1) of degree $2p^i - 2, equipped with the coproduct

\psi(\xi_r) = \sum_{i=0}^r \xi_{r-i}^{p^i} \otimes \xi_i, \quad \psi(\tau_r) = \tau_r \otimes 1 + \sum_{i=0}^r \xi_{r-i}^{p^i} \otimes \tau_i

for r \geq 0 (with the sum over i such that the indices are nonnegative). For p=2, the exterior factor is absent, and \overline{\mathcal{A}}_2 = P(\xi_1, \xi_2, \dots) with degrees $2^i - 1 and coproduct \psi(\xi_r) = \sum_{i=0}^r \xi_{r-i}^{2^i} \otimes \xi_i. The product and coproduct on \mathcal{A}_p are the linear duals of the coproduct and product on \overline{\mathcal{A}}_p, respectively, yielding a Hopf algebra structure independent of topological input.^[18] This dual description aligns with the functional realization of \mathcal{A}_p via its action on the polynomial ring \mathbb{F}_p, where \overline{\mathcal{A}}_p acts by multiplication: the generator \xi_1 corresponds to multiplication by x^p, and higher generators follow from the coproduct structure, allowing explicit computation of operations as power series expansions.^[18] Such an algebraic model facilitates computational methods in the Steenrod algebra without recourse to topological spectra or mapping spaces, enabling efficient handling of relations and bases through linear algebra over \mathbb{F}_p.

Algebraic Structure

Hopf Algebra Structure

The Steenrod algebra \mathcal{A}_p over the prime field \mathbb{F}_p possesses a rich Hopf algebra structure, which endows it with both an algebra and a coalgebra structure compatible via an antipode. The multiplication in \mathcal{A}_p is given by composition of cohomology operations, with the unit map sending the generator of \mathbb{F}_p to the identity operation of degree zero. The counit \epsilon: \mathcal{A}_p \to \mathbb{F}_p is the augmentation that projects onto the degree-zero component, sending all positive-degree operations to zero. This structure ensures that \mathcal{A}_p acts as a bialgebra on the cohomology of spaces, facilitating the study of operations on products via the diagonal map.^[2] The coproduct \psi: \mathcal{A}_p \to \mathcal{A}_p \otimes \mathcal{A}_p is a graded algebra homomorphism that encodes the compatibility with the cup product in cohomology. For p = 2, it is defined on the generators by

\psi(\mathrm{Sq}^i) = \sum_{j=0}^i \mathrm{Sq}^j \otimes \mathrm{Sq}^{i-j},

and extends multiplicatively to the entire algebra. For odd primes p, the coproduct on the power operations is similarly

\psi(P^i) = \sum_{j=0}^i P^j \otimes P^{i-j},

with the Bockstein \beta being primitive: \psi(\beta) = \beta \otimes 1 + 1 \otimes \beta. This coproduct is commutative and coassociative, making \mathcal{A}_p a cocommutative Hopf algebra over \mathbb{F}_p.^[1]^[2] The coproduct induces the Cartan formula, which describes the action of operations on cup products. For classes x \in H^*(X; \mathbb{F}_p) and y \in H^*(Y; \mathbb{F}_p), an operation \alpha \in \mathcal{A}_p acts via

\alpha(xy) = \sum \alpha_{(1)}(x) \, \alpha_{(2)}(y),

where \sum \alpha_{(1)} \otimes \alpha_{(2)} = \psi(\alpha) in Sweedler notation. For the specific generators at p=2,

\mathrm{Sq}^i(xy) = \sum_{j=0}^i \mathrm{Sq}^j(x) \, \mathrm{Sq}^{i-j}(y),

and for odd p,

P^i(xy) = \sum_{j=0}^i P^j(x) \, P^{i-j}(y),

with a similar Leibniz rule for the Bockstein involving a sign. This formula is fundamental for computing Steenrod operations on products and underpins applications in algebraic topology.^[1] The antipode \chi: \mathcal{A}_p \to \mathcal{A}_p is an involutive graded antiautomorphism (\chi^2 = \mathrm{id}) that provides the inverse under convolution, completing the Hopf algebra structure. For p=2, it is defined recursively by \chi(\mathrm{Sq}^0) = \mathrm{Sq}^0 and, for n > 0,

\sum_{k=0}^n \mathrm{Sq}^k \chi(\mathrm{Sq}^{n-k}) = 0.

Computations are performed using this recursion together with the Adem relations. For odd p, an analogous recursive definition holds, with explicit formulas available in the Milnor basis involving signs such as (-1)^{i+1} for terms on P^i. On admissible monomials, the antipode includes signs based on the length of the sequence. The antipode ensures the existence of inverses in the convolution algebra and is crucial for duality properties.^[2] The dual Hopf algebra \mathcal{A}_p^* admits a simple description as an exterior algebra on generators \tau_i ( i \geq 0, degree $2p^i - 1) tensored with a polynomial algebra on \xi_i ( i \geq 1, degree $2p^i - 2): \mathcal{A}_p^* \cong E(\tau_i \mid i \geq 0) \otimes P(\xi_i \mid i \geq 1). The \tau_i are primitive under the dual coproduct, reflecting the cocommutative nature of \mathcal{A}_p. This explicit form facilitates computations and reveals the combinatorial underpinnings of the Steenrod algebra.^[2]

Bases and Antiautomorphisms

The admissible basis for the mod 2 Steenrod algebra, also known as the Serre-Cartan basis, consists of monomials of the form \mathrm{Sq}^{i_1} \mathrm{Sq}^{i_2} \cdots \mathrm{Sq}^{i_r}, where the indices satisfy i_j \ge 2 i_{j+1} for $1 \le j \le r-1 and i_r \ge 0. The degree of such a monomial is \sum_{j=1}^r i_j. These monomials form an additive basis for the algebra as an \mathbb{F}_2-vector space.^[19]^[2] The dimension of the degree n component A^n is the number of admissible monomials of total degree n. For general primes p, the analogous admissible basis uses the condition i_j \ge p i_{j+1}, and the dimension of A_p^n is p^{\mu(n)}, where \mu(n) is the number of non-zero digits in the base-p expansion of n.^[2]^[20] Another standard basis is the Milnor basis, introduced by J. W. Milnor. For p = 2, it consists of elements Q(i_1, i_2, \dots , i_r), dual to the monomials \xi_{i_1} \xi_{i_2} \cdots \xi_{i_r} in the dual algebra \mathbb{F}_2[\xi_i \mid i \ge 1], with degree \sum_{k=1}^r (2^{i_k} - 1). For odd primes p, the dual algebra is \mathbb{F}_p[\xi_i, \tau_i \mid i \ge 0] with \tau_0 included, and the Milnor basis is the dual monomials in these generators. The multiplication in the Milnor basis is given by combinatorial formulas involving multinomial coefficients modulo p. The change of basis matrix between the admissible and Milnor bases is upper triangular with entries that are binomial coefficients modulo p, as computed by H. R. Margolis.^[2]^[19] The canonical antiautomorphism \chi, first studied by R. Thom, is defined recursively by \chi(\mathrm{Sq}^0) = \mathrm{Sq}^0 and, for n > 0,

\sum_{k=0}^n \mathrm{Sq}^k \chi(\mathrm{Sq}^{n-k}) = 0.

The recursion is used in practice with the Adem relations to compute it. This \chi is an antiautomorphism of the algebra, satisfying \chi(ab) = \chi(b) \chi(a), and it is a Hopf antiautomorphism, compatible with the coproduct via \psi(\chi(a)) = \chi \otimes \chi \, \psi^{\mathrm{op}}(a), where \psi^{\mathrm{op}} is the opposite coproduct. It plays a key role in establishing dualities between cohomology operations and stable homotopy groups.^[2]^[20]

Finite Sub-Hopf Algebras

Finite sub-Hopf algebras of the Steenrod algebra A_p at prime p are finite-dimensional Hopf subalgebras that provide a filtration of A_p by increasing chains of subalgebras, facilitating computations in low degrees. A key family is the sub-Hopf algebras E(n) for n \geq 0, defined as the subalgebra generated by elements Q_0, \dots, Q_{n-1}, forming an exterior algebra \Lambda(Q_0, \dots, Q_{n-1}) where the Q_i have degrees $2p^i - 1. In general, E(n) has dimension p^n. For p=2, Q_0 = \mathrm{Sq}^1, and higher Q_i are defined via the change of basis to make E(n) a Hopf subalgebra.^[21] Another important family is the sub-Hopf algebras A(n), which consist of all elements of A_p in internal degrees less than p^n. These algebras have a basis given by the admissible monomials in the Serre-Cartan basis (products of Steenrod operations satisfying the admissibility condition, such as i_j \geq p i_{j+1} for odd p, or i_j \geq 2 i_{j+1} for p=2) of degree less than p^n. The dimension of A(n) is p^{n(n+1)/2}, matching the structure in certain computational contexts but differing from E(n) as A(n) includes non-primitive elements.^[21]^[22] The direct limit E(\infty) = \bigcup_n E(n) is the exterior Hopf algebra on the infinite set of all such generators of A_p, while A(\infty) = \bigcup_n A(n) recovers the full Steenrod algebra A_p. For p=2, a concrete example is E(2), the exterior algebra on Q_0 = \mathrm{Sq}^1 (degree 1) and Q_1 = \mathrm{Sq}^3 + \mathrm{Sq}^2 \mathrm{Sq}^1 (degree 3), of dimension 4 with basis $1, Q_0, Q_1, Q_0 Q_1. This aligns with the structure in low degrees.^[2]^[22]

Computations and Examples

On Complex Projective Spaces

The cohomology of the infinite complex projective space \mathbb{CP}^\infty with \mathbb{F}_p-coefficients is the polynomial algebra \mathbb{F}_p, where x \in H^2(\mathbb{CP}^\infty; \mathbb{F}_p) generates the ring as an algebra over \mathbb{F}_p.^[8] This structure makes \mathbb{CP}^\infty a fundamental example for studying the module structure of cohomology rings over the Steenrod algebra \mathcal{A}_p, as the even-dimensional cell structure leads to a straightforward action of the operations.^[8] For p=2, the odd-indexed Steenrod squares \Sq^{2k+1} vanish on H^*(\mathbb{CP}^\infty; \mathbb{F}_2) due to degree reasons, while the even-indexed squares act via the formula \Sq^{2k}(x^m) = \binom{m}{k} x^{m+k} \mod 2 for k \leq m and $0 otherwise.^[8] Representative computations include \Sq^2(x) = x^2, \Sq^2(x^2) = 0 (since \binom{2}{1} \equiv 0 \mod 2), and \Sq^4(x^2) = x^4 (since \binom{2}{2} = 1). The higher operations can be decomposed using the Adem relations to express them in terms of products involving \Sq^2; for instance, relations like \Sq^a \Sq^b = \sum_{j \geq 0} \binom{b-j-1}{a-2j} \Sq^{a+b-j} \Sq^j when $0 < a < 2b allow reduction to a basis of admissible monomials in the generators.^[8] For odd primes p, the reduced power operations P^k (of degree k(p-1)) and the Bockstein \beta (of degree 1) generate \mathcal{A}_p, with \beta acting trivially on the even-degree ring. The action is given by P^k(x^m) = \binom{m}{k} x^{m + k \frac{p-1}{2}} \mod p, where the binomial coefficient is computed modulo p.^[1] The coefficient \binom{m}{k} \mod p vanishes unless the base-p digits of k are componentwise at most those of m, by Lucas' theorem; for example, with p=3, P^1(x) = x^2 (since \binom{1}{1} = 1) and P^1(x^2) = 2x^{3} \equiv -x^{3} \mod 3 (since \binom{2}{1} = 2). Adem relations for odd p, such as P^a P^b = \sum_j \binom{(p-1)(b-j)-1}{a - p j} P^{a+b-j} P^j for a < p b, along with \beta^2 = 0 and mixed relations, enable expressing higher powers in terms of primitives like P^1 and \beta.^[8] The elements annihilated by all positive-dimensional operations (the positive-degree part of the kernel of the augmentation ideal action) are spanned by the monomials x^{p^r} for r \geq 1, as \binom{p^r}{k} \equiv 0 \mod p for $0 < k < p^r by properties of binomial coefficients modulo p (via Lucas' theorem).^[8] The subring generated by these invariant monomials is \mathbb{F}_p[x^{p}, x^{p^2}, x^{p^3}, \dots], which is fixed setwise by the Steenrod operations. This structure highlights the hierarchical nature of the module, with each "layer" corresponding to p-power exponents.^[8] As the Eilenberg–MacLane space K(\mathbb{Z}, 2), \mathbb{CP}^\infty models complex K-theory in low dimensions, and the Steenrod action encodes power operations that relate ordinary cohomology to generalized theories like complex cobordism, where the operations correspond to transfers and norms in the bordism ring.^[8]

On Real Projective Spaces

The cohomology ring of the infinite real projective space \mathbb{RP}^\infty with \mathbb{F}_2 coefficients is the polynomial algebra \mathbb{F}_2, where w \in H^1(\mathbb{RP}^\infty; \mathbb{F}_2) is the degree-1 generator corresponding to the fundamental class.^[6]^[8] This structure arises as the direct limit of the finite-dimensional cases, yielding \mathbb{F}_2 in every nonnegative degree, with the ring multiplication given by w^k \cdot w^l = w^{k+l}.^[6] In contrast to the even-degree polynomial algebra on complex projective spaces, the odd-degree generator here leads to a commutative polynomial ring that fills both even and odd dimensions uniformly.^[8] The action of the Steenrod squares on this cohomology is determined by their behavior on the generator: \mathrm{Sq}^0(w) = w, \mathrm{Sq}^1(w) = w^2, and \mathrm{Sq}^i(w) = 0 for i > 1, reflecting the degree increase and the Cartan formula for products.^[6]^[1] Extending to powers via the Cartan formula and the Leibniz rule modulo 2, the general action is given by

\mathrm{Sq}^n(w^k) = \binom{k}{n} w^{k+n} \pmod{2},

where the binomial coefficient \binom{k}{n} \pmod{2} is nonzero precisely when the binary expansion of n is contained in that of k (by Lucas' theorem).^[6]^[8] This formula produces nontrivial relations among powers, such as \mathrm{Sq}^{2^{m-1}}(w^{2^{m-1}}) = w^{2^m}, linking lower powers to higher ones.^[1] The subring of invariants under the full Steenrod algebra action consists of the classes such that the action maps the subring to itself setwise, which for \mathbb{RP}^\infty is generated by the monomials w^{2^k} for k \geq 1; these correspond to the Stiefel-Whitney classes w_{2^k} in the classifying space context.^[8]^[1] Thus, the invariant subring is \mathbb{F}_2[w^2, w^4, w^8, \dots], emphasizing the role of powers of 2 in the binary structure.^[6] For the finite real projective space \mathbb{RP}^n, the cohomology ring is the truncated polynomial \mathbb{F}_2 / (w^{n+1}), where H^k(\mathbb{RP}^n; \mathbb{F}_2) = \mathbb{F}_2 for $0 \leq k \leq n and zero otherwise, with the same generator w \in H^1.^[6] The Steenrod square action mirrors the infinite case up to degree n, via the same binomial formula \mathrm{Sq}^n(w^k) = \binom{k}{n} w^{k+n} \pmod{2} provided k + n \leq n, but truncates beyond degree n, leading to additional relations like w^{n+1} = 0.^[8] This truncation distinguishes finite approximations from the untruncated infinite structure, where no such cutoff occurs, allowing the full polynomial growth and invariant subring to manifest without boundary effects.^[1]

Generalizations

To Generalized Cohomology Theories

In the context of a generalized cohomology theory represented by an Ω-spectrum E, the analogue of the Steenrod algebra is the graded ring [E, E]_* of stable homotopy classes of self-maps of E in the stable homotopy category.^[23] This ring encodes the natural transformations of the functor E^* to itself, generalizing the action of cohomology operations on spaces.^[23] A prominent example arises in complex K-theory, represented by the spectrum KU. Here, the corresponding Steenrod algebra [KU, KU]_* is concentrated in even degrees and generated by the Adams operations \psi^k for k \geq 1, each of degree 0. These operations are ring homomorphisms satisfying \psi^k \circ \psi^l = \psi^{kl} and \psi^1 = \mathrm{id}, forming a polynomial ring structure on the reduced operations \psi^k - k over \mathbb{Z}. The Adams operations arise from the action of exterior powers in the representation ring and provide power operations analogous to the Steenrod powers in ordinary cohomology. For complex cobordism, represented by the spectrum MU, the Steenrod algebra is the Hopf algebroid (MU_*(MU), MU_*) known as the Landweber-Novikov algebra.^[23] This algebra is generated over the Lazard ring MU_* = \mathbb{L} by operations Q(\lambda) corresponding to partitions \lambda, with additional structure from the \beta-Bockstein operations associated to the cofiber sequences in the p-series of the universal formal group law. The Landweber-Novikov operations Q(\lambda) act on MU_*(X) via a comodule structure, enabling computations in the Adams-Novikov spectral sequence.^[23] The axiomatic characterization adapts the classical properties to this setting: operations exhibit stability, mapping E^n(X) to E^{n+k}(X) for large n, and satisfy a Cartan formula for the action on products, \alpha(x \cdot y) = \sum \alpha_i(x) \cdot \alpha_i'(y), reflecting the Hopf algebra structure.^[23] However, the structure is richer due to the complex coefficient rings, incorporating formal group laws and leading to more intricate relations like Adem-type rules derived from comodule algebra.^[23] The ordinary mod-p Steenrod algebra A_p embeds into these generalized versions via the change-of-rings isomorphism induced by the unit map H\mathbb{F}_p \to E, which identifies operations on ordinary cohomology with those on E-cohomology after tensoring with E_*.^[23] This embedding preserves key properties, such as the action on Eilenberg-MacLane spaces, and facilitates comparisons between theories.^[23]

Motivic and Equivariant Versions

The motivic Steenrod algebra arises as the bigraded algebra of bistable cohomology operations acting on the mod p motivic cohomology of smooth schemes over a base scheme S.^[24] For p=2, it is generated by operations Sq^{a,b} where a \geq b \geq 0, with bidegree (a,b) in the cohomological and weight gradings, respectively. For odd primes p (over bases of characteristic zero), it coincides with the classical Steenrod algebra \mathcal{A}_p, generated by P^i (i \geq 1) and \beta, with bidegrees (2i(p-1), i(p-1)) for P^i and (1,0) for \beta. In positive characteristic, the structure is analogous but generated by operations P^{a,b} and \beta with adjusted bidegrees, such as (2a(p-1), a(p-1)) for P^{a,b}.^[24] These generators extend the classical Steenrod operations to the motivic setting, where the weight grading reflects the additional structure from algebraic cycles.^[25] The relations among these generators are governed by motivic analogs of the Adem relations, which express higher operations as polynomials in lower ones, ensuring the algebra's presentation mirrors the classical case while incorporating the bigrading. Furthermore, the motivic Steenrod algebra acts on the motivic cohomology of the base point, which for H^{2i,i}(\mathrm{Spec}\, k, \mathbb{Z}/p\mathbb{Z}) recovers the p-completed Milnor K-theory of the field k, linking algebraic K-theory to stable operations in this context.^[26] This action provides a bridge between topological invariants and arithmetic geometry. In the equivariant setting, the Steenrod algebra generalizes to finite groups G, comprising the stable EG-equivariant cohomology operations on H_G^*(X; \mathbb{F}_p) for G-spaces X.^[27] These operations form a Hopf algebra over the equivariant coefficient ring, extending the classical structure to incorporate group actions via Borel or geometric constructions.^[27] Recent advancements include computations of the \mathbb{R}-motivic Steenrod algebra's action on Spanier-Whitehead duals of finite complexes, revealing how dual cohomology modules recover original structures as modules over this algebra.^[28] Post-2020 equivariant generalizations employ geometric fixed point spectra to compute dual Steenrod algebras for cyclic groups C_p, yielding explicit module decompositions over constant Mackey functors.^[29]

Applications

Adams Spectral Sequence

The Adams spectral sequence provides a method to compute the stable homotopy groups of spheres using the cohomology of the sphere spectrum and the action of the Steenrod algebra. In the mod 2 case, the E_2-term is given by the Ext groups over the Steenrod algebra A:

E_2^{s,t} = \Ext_A^{s,t}(\mathbb{F}_2, H^*(S^0; \mathbb{F}_2)) \Rightarrow \pi_{t-s}(S^0),

where the convergence is to a filtration on the 2-primary component of the stable stems, with the spectral sequence arising from a minimal resolution of the trivial module via the cobar construction on the dual Steenrod algebra.^[17]^[2] This setup leverages the Hopf algebra structure of A to translate cohomological data into homotopical information, resolving the problem of computing homotopy groups from cohomology by accounting for the action of Steenrod operations on the Eilenberg-MacLane spectrum resolution of the sphere.^[17] Differentials in the Adams spectral sequence originate from secondary cohomology operations and higher Toda brackets, which detect obstructions to extending primary operations and encode relations in the homotopy groups. For instance, at the prime 2, these differentials arise from the unstable relations in the Steenrod algebra and the geometry of the attaching maps in the cell structure of spheres, often computed using the cobar differential induced by the coproduct in the dual Steenrod algebra.^[30] The role of the mod p Steenrod algebra A_p is central, as it acts on mod p cohomology to filter the resolution, effectively bridging cohomology to homotopy by quotienting out the action and using Ext to capture the derived functor information. Computations of these Ext groups are facilitated by the May spectral sequence, which filters A_p by powers of its generators to converge to the E_2-page.^[31] Key applications include the resolution of the Hopf invariant one problem, where Adams used the Adem relations to show that elements of Hopf invariant one can only exist when the dimension n of the base sphere is a power of 2, with existence confirmed only in dimensions 3 and 7.^[30] Similarly, the image of the J-homomorphism, which maps orthogonal representations to homotopy classes, is determined via the Adams spectral sequence by identifying permanent cycles corresponding to generators in the Ext groups that survive to the E_∞-page.^[32] The connection to the dual Steenrod algebra emphasizes its use as a comodule algebra, where the cobar complex on the dual computes the requisite Ext groups as the cohomology of this comodule structure over the trivial comodule.^[2]

Modern Uses in Topology

In recent developments within topological data analysis, persistent Steenrod modules have emerged as a powerful tool for enhancing the invariants derived from mod 2 persistent cohomology. Introduced in 2022, these modules leverage computations of Steenrod squares to produce Sq^k-barcodes, which capture the action of Steenrod operations on filtrations and detect topological features finer than standard Betti numbers, such as non-trivial interactions in data structures like point clouds or simplicial complexes.^[33] This approach provides an algorithmic pipeline for computing these invariants, enabling more discriminative analysis in applications ranging from shape recognition to sensor network topology.^[34] Efforts to realize the Steenrod algebra as an enveloping algebra have faced significant obstacles, particularly in the context of Lie superalgebras. A 2025 study demonstrates that the Steenrod algebra cannot be realized as the enveloping algebra of any Lie superalgebra, highlighting structural incompatibilities that arise from the algebra's coproduct and grading.^[35] This impossibility underscores the unique Hopf algebra properties of the Steenrod algebra and motivates modifications to realization frameworks, such as adjusted definitions for superalgebra extensions. Graph-theoretic interpretations of the dual Steenrod algebra have also seen advancements, with 2025 work extending constructions to mod p settings and establishing connectedness criteria for graphs associated to monomials in certain quotients.^[36] These graphs reveal connectivity patterns that reflect the algebraic structure, aiding in the study of subalgebras and their quotients, with implications for understanding the dual's combinatorial properties. Beyond classical topology, the Steenrod algebra finds applications in equivariant chromatic homotopy theory, where equivariant versions facilitate computations of stable homotopy groups via spectral sequences and power operations on the dual algebra. In topological data analysis, Steenrod operations detect persistent features like higher-dimensional holes and cycles, improving feature extraction in noisy datasets.^[34] Motivic extensions further bridge to algebraic geometry, with recent constructions of power operations on mod-p motivic cohomology using syntomic refinements to analyze schemes over finite fields.^[37] These developments fill gaps in classical applications by integrating the algebra into modern computational and geometric contexts.

References

[1]
[PDF] COHOMOLOGY OPERATIONS
Chapter I presents the squares axiomatically, all of their main properties are assumed. In Chapters II, III, and IV, further properties are developed, and the ...
[2]
[PDF] The Steenrod Algebra and Its Dual
The Steenrod algebra has a canonical anti-automorphism which was first studied by R. ... Next we will study certain subalgebras of the Steenrod algebra. Adem.
[3]
Products of Cocycles and Extensions of Mappings - jstor
BY N. E. STEENROD. (Received May 6, 1946). 1. Introduction. The Brouwer concept of ... Passing to cohomology classes (and reducing mod 2 when p - i is even),.
[4]
On the structure and applications of the steenrod algebra
Article. On the structure and applications of the steenrod algebra. Published: December 1958. Volume 32, pages 180–214, (1958); Cite this article. Download PDF.
[5]
[PDF] Foundations of Algebraic Topology
The principal contribution of this book is an axiomatic approach to the part of algebraic topology called homology theory. It is the oldest.
[6]
[PDF] Algebraic Topology - Cornell Mathematics
This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in ...Missing: Foundations | Show results with:Foundations
[7]
[PDF] HISTORY OF HOMOLOGICAL ALGEBRA Charles A. Weibel ...
Homological algebra had its origins in the 19th century, via the work of Riemann. (1857) and Betti (1871) on “homology numbers,” and the rigorous ...
[8]
[PDF] Cohomology Operations and Applications in Homotopy Theory
In the past two decades, cohomology operations have been the center of a major area of activity in algebraic topology. This technique for supplementing.
[9]
[PDF] A general algebraic approach to steenrod operations
We emphasize that this is an expository paper. Although a number of new re- sults and new proofs of old results are scattered throughout, the only real claim to.
[10]
[PDF] The Steenrod algebra - Purdue Math
Jan 25, 2016 · ask for stable cohomology operations which are stable. A degree k stable cohomology operation is a natural transformation. HnX → Hn+kX for ...
[11]
ON THE ACTION OF STEENROD SQUARES ON POLYNOMIAL ...
(Z/2Z) x • □ • x (Z/2Z) (s factors). The mod-2 Steenrod algebra A acts on. Ps according to well-known rules. If A C A denotes the augmentation ideal,.
[12]
[PDF] Steenrod Algebra Suminar: Construction of Steenrod Operations
Jun 2, 2011 · We will focus on the so-called Steenrod operations. They look a little different depending on whether p is odd or 2, so we list the properties ...
[13]
On the Adem relations | Mathematical Proceedings of the ...
Oct 24, 2008 · (1)Adem, J., The relations on Steenrod powers of cohomology classes. Algebraic geometry and topology, ed. Fox, R. H. et al.Missing: original | Show results with:original
[14]
[PDF] An algebraic introduction to the Steenrod algebra - arXiv
Nov 14, 2007 · The purpose of these notes is to provide an introduction to the Steenrod algebra in an algebraic manner avoiding any use of cohomology ...
[15]
[PDF] PROBLEMS IN THE STEENROD ALGEBRA
This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in ...
[16]
[PDF] THE STEENROD ALGEBRA AND ITS DUAL 1. Cohomology and Eile
Feb 20, 2023 · It is commutative if A and B are commutative, in which case it is the coproduct. (= categorical sum) of A and B in the category of commutative k ...<|control11|><|separator|>
[17]
[PDF] On the Structure and Applications of the STEENROD Algebra.
Apr 29, 2012 · The integer j must be odd (if p > 2). The tmncated divided polynomial algebra P(i,j ;k) (over Zp) has a Zp-base.
[18]
The Steenrod Algebra and Its Dual - jstor
The Steenrod algebra has a canonical anti-automorphism which was first studied by R. Thom. This anti-automorphism is computed in ?7.
[19]
https://arxiv.org/pdf/0903.4997.pdf
[20]
[PDF] on the anti-automorphism of the Steenrod algebra - MIT Mathematics
To prove this formula, note that the defining identity for binomial coefficients implies the case m = 1, and also that both sides satisfy the recursion C(l,m,n) ...
[21]
Monomial Bases in the Mod-<Emphasis Type="Italic">p</Emphasis ...
MONOMIAL BASES IN THE MOD-p STEENROD ALGEBRA. , Izmir. (Received October 22, 2002). Abstract. In this paper we study sets of some special monomials which form ...
[22]
The Steenrod algebra - SageMath Documentation
If the two elements are represented in the Serre-Cartan basis, then multiply them using Adem relations (also implemented in sage.algebras.steenrod.
[23]
[PDF] Sub-Hopf algebras of the Steenrod algebra and the Singer transfer
Let A denote the mod 2 Steenrod algebra (see Steenrod and Epstein [28]). The problem of computing its cohomology H∗,∗(A) is of great importance in algebraic ...
[24]
[PDF] Complex Cobordism and Stable Homotopy Groups of Spheres
May 1, 2025 · Page 1. Complex Cobordism and. Stable Homotopy Groups of Spheres. Douglas C. Ravenel ... Ravenel–Wilson Hopf ring. The unstable Adams–Novikov ...
[25]
[1305.5690] The motivic Steenrod algebra in positive characteristic
May 24, 2013 · We show that the algebra of bistable operations in the mod l motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations.
[26]
[PDF] REDUCED POWER OPERATIONS IN MOTIVIC COHOMOLOGY
The proof of the following theorem which provides Adem relations for odd l follows the same line of arguments as the proof of the corresponding topological fact.
[27]
The equivariant Steenrod algebra - ScienceDirect.com
This paper presents a generalization of the mod [rgr] Steenrod algebra A∗, to G-equivariant cohomology theory for a finite group G. The coefficient ring ...
[28]
On the Steenrod module structure of ℝ-motivic Spanier-Whitehead ...
Oct 30, 2024 · In this paper, we describe how to recover the R-motivic cohomology of the Spanier–Whitehead dual DX of an R- motivic finite complex X, as an AR- ...
[29]
[2103.16006] On the $C_p$-equivariant dual Steenrod algebra - arXiv
Mar 30, 2021 · We compute the C_p-equivariant dual Steenrod algebras associated to the constant Mackey functors \underline{\mathbb{F}}_p and \underline{\mathbb{Z}}_{(p)}, as ...
[30]
[PDF] On the Non-Existence of Elements of Hopf Invariant One
Results. It is the object of this paper to prove a theorem in homo- topy-theory, which follows as Theorem 1.1.1. In stating it, we use one definition.
[31]
[PDF] A general algebraic approach to Steenrod operations
For example, there are Steenrod operations in the cohomology of simplicial restricted Lie algebras, in the cohomology of cocom- mutative Hopf algebras, and in ...
[32]
On the groups J(X)—IV - ScienceDirect.com
Adams. On the groups J(X)—I. Topology, 2 (1963), pp. 181-195. View PDFView articleView in Scopus Google Scholar. 4. J.F. Adams. On the groups J(X)—II.
[33]
Persistence Steenrod modules | Journal of Applied and ...
May 29, 2022 · ... p, i) coproducts for simplicial and cubical chains using the ... where \lfloor - \rfloor denotes the integer part function and the binomial ...
[34]
[1812.05031] Persistence Steenrod modules - arXiv
We present a complete algorithmic pipeline for their computation and illustrate their real-world applicability using the space of conformations ...Missing: motivic | Show results with:motivic
[35]
[2509.09443] On realisations of the Steenrod algebras - arXiv
Sep 11, 2025 · The Steenrod algebra can not be realised as an enveloping of any Lie superalgebra. We list several problems that suggest a need to modify the ...
[36]
[2508.17041] Graphs arising from the dual Steenrod algebra - arXiv
Aug 23, 2025 · We also give graph theoretic interpretations of algebraic structures such as the coproduct and antipode arising from the Hopf algebra structure ...
[37]
[2506.05585] Motivic Steenrod operations at the characteristic via ...
Jun 5, 2025 · Our operations satisfy the expected properties (naturality, Adem relations, and the Cartan formula) for all bidegrees, generalizing previous ...