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Cohomology Groups of the Dual Steenrod Algebra Cohomology Groups of the Dual Steenrod Algebra Ryan Kim∗ Mentor : Sanath Devalapurkar MIT PRIMES-USA ∗Thomas Jefferson High School for Science and Technology May 19, 2018 Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 1 / 21 Example Integers Z with standard + form a group • 0 is the identity element and inverses are additive inverses Abstract Algebra Fundamentals Groups A group is a set of elements with an \addition" operation + • + is associative, with an identity element and inverses Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 2 / 21 Abstract Algebra Fundamentals Groups A group is a set of elements with an \addition" operation + • + is associative, with an identity element and inverses Example Integers Z with standard + form a group • 0 is the identity element and inverses are additive inverses Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 2 / 21 Abstract Algebra Fundamentals Rings A ring can be thought of as a group with an additional operation and more restrictions • + now commutative, also a \multiplication" operation × • × must be associative, distribute over +, and have an identity Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 3 / 21 Abstract Algebra Fundamentals Example Set of 2 by 2 matrices with real entries M2(R) is a ring, with addition and matrix multiplication • Multiplication is not commutative, multiplicative inverse doesn't always exist a11 a12 b11 b12 a11 + b11 a12 + b12 • + = a21 a22 b21 b22 a21 + b21 a22 + b22 a11 a12 b11 b12 a11b11 + a12b12 a11b12 + a12b22 • = a21 a22 b21 b22 a21b11 + a22b21 a21b12 + a22b22 Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 4 / 21 Example Set of integers modulo p (a prime) with typical addition and multiplication form a field Fp −1 • 2 ≡ 4 (mod 7) • 0 does not have multiplicative inverse Abstract Algebra Fundamentals Fields A field can be thought of as a ring with a few more restrictions on the multiplication operation • Commutative multiplication, a multiplicative identity, and multiplicative inverses (except for the addition identity) Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 5 / 21 Abstract Algebra Fundamentals Fields A field can be thought of as a ring with a few more restrictions on the multiplication operation • Commutative multiplication, a multiplicative identity, and multiplicative inverses (except for the addition identity) Example Set of integers modulo p (a prime) with typical addition and multiplication form a field Fp −1 • 2 ≡ 4 (mod 7) • 0 does not have multiplicative inverse Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 5 / 21 Example The complex numbers are an algebra over the reals • (a + bi) + (c + di) = (a + b) + (c + d)i • (a + bi)(c + di) = (ac − bd) + (ad + bc)i • r(a + bi) = (ra) + (rb)i, for r ∈ R Algebras over Fields Algebras An algebra is a set of elements with addition, multiplication, and scalar multiplication over a field Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 6 / 21 Algebras over Fields Algebras An algebra is a set of elements with addition, multiplication, and scalar multiplication over a field Example The complex numbers are an algebra over the reals • (a + bi) + (c + di) = (a + b) + (c + d)i • (a + bi)(c + di) = (ac − bd) + (ad + bc)i • r(a + bi) = (ra) + (rb)i, for r ∈ R Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 6 / 21 Modules Modules A module is like an algebra. However, modules scale over rings instead of fields, and do not require the bilinear product. Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 7 / 21 dual Steenrod Algebra i A∗ = Z~2[ξ1; ξ2;:::] where each ξi has degree 2 − 1 modulo 2 dual Steenrod algebra • Consider the Steenrod algebra given by p = 2 (from algebraic topology) • Obtain dual algebra by considering linear maps from the algebra to the field it is considered over, or Z~2 • The modulo 2 dual Steenrod Algebra, denoted by A∗, is a polynomial ring (Milnor) Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 8 / 21 modulo 2 dual Steenrod algebra • Consider the Steenrod algebra given by p = 2 (from algebraic topology) • Obtain dual algebra by considering linear maps from the algebra to the field it is considered over, or Z~2 • The modulo 2 dual Steenrod Algebra, denoted by A∗, is a polynomial ring (Milnor) dual Steenrod Algebra i A∗ = Z~2[ξ1; ξ2;:::] where each ξi has degree 2 − 1 Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 8 / 21 Conjugation Action k 2i Q χ(ξi) ⋅ ξk−i = 0 i=0 22 21 20 • Example: χ(ξ2) ⋅ ξ0 + χ(ξ1) ⋅ ξ1 + χ(ξ0) ⋅ ξ2 = 0 • χ respects multiplication and addition: • χ(ab) = χ(a)χ(b) • χ(a + b) = χ(a) + χ(b) • χ preserves degree 2 • χ = 0 Action of Z 2 on ∗ • Denote nontrivial~ A element of Z~2 by χ • Define the canonical conjugation action of Z~2 on A∗ inductively: Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 9 / 21 22 21 20 • Example: χ(ξ2) ⋅ ξ0 + χ(ξ1) ⋅ ξ1 + χ(ξ0) ⋅ ξ2 = 0 • χ respects multiplication and addition: • χ(ab) = χ(a)χ(b) • χ(a + b) = χ(a) + χ(b) • χ preserves degree 2 • χ = 0 Action of Z 2 on ∗ • Denote nontrivial~ A element of Z~2 by χ • Define the canonical conjugation action of Z~2 on A∗ inductively: Conjugation Action k 2i Q χ(ξi) ⋅ ξk−i = 0 i=0 Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 9 / 21 • χ preserves degree 2 • χ = 0 Action of Z 2 on ∗ • Denote nontrivial~ A element of Z~2 by χ • Define the canonical conjugation action of Z~2 on A∗ inductively: Conjugation Action k 2i Q χ(ξi) ⋅ ξk−i = 0 i=0 22 21 20 • Example: χ(ξ2) ⋅ ξ0 + χ(ξ1) ⋅ ξ1 + χ(ξ0) ⋅ ξ2 = 0 • χ respects multiplication and addition: • χ(ab) = χ(a)χ(b) • χ(a + b) = χ(a) + χ(b) Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 9 / 21 Action of Z 2 on ∗ • Denote nontrivial~ A element of Z~2 by χ • Define the canonical conjugation action of Z~2 on A∗ inductively: Conjugation Action k 2i Q χ(ξi) ⋅ ξk−i = 0 i=0 22 21 20 • Example: χ(ξ2) ⋅ ξ0 + χ(ξ1) ⋅ ξ1 + χ(ξ0) ⋅ ξ2 = 0 • χ respects multiplication and addition: • χ(ab) = χ(a)χ(b) • χ(a + b) = χ(a) + χ(b) • χ preserves degree 2 • χ = 0 Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 9 / 21 Action of Z 2 on ∗ ~ A Some small values of χ(ξi): (Note that ξ0 = 1) Computed Values χ(1) = 1 χ(ξ1) = ξ1 3 χ(ξ2) = ξ2 + ξ1 2 4 7 χ(ξ3) = ξ3 + ξ1ξ2 + ξ1ξ2 + ξ1 2 8 5 3 4 9 2 12 15 χ(ξ4) = ξ4 + ξ1ξ3 + ξ1ξ3 + ξ2 + ξ1ξ2 + ξ1ξ2 + ξ1 ξ2 + ξ1 2 3 1 Homogenous: deg ξ2 = 2 − 1 = 3, deg ξ1 = 3(2 − 1) = 3 Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 10 / 21 Homological Algebra • Associate sequences of algebraic objects with other algebraic objects • Example: Homology groups Hn(X) • Elucidate information about "holes" in topological spaces Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 11 / 21 Group Cohomology • View A∗ as a module over Z~2 • Let Z~2 act on A∗ by the conjugation action n • The cohomology groups H (G; M) of a module M and its G-action elucidate information about the action • Goal: compute cohomology groups of this action Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 12 / 21 Fixed points 0 χ • Zeroth cohomology group H (Z~2; A∗) is subalgebra A∗ invariant under the action χ • We would like to compute these fixed points of χ Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 13 / 21 Summary of work • Some elements are clearly invariant under χ: Invariant Elements • 1, ξ1 • χ() for any • χ() + for any • However, these elements do not span the fixed point space! 3 2 9 • Example: ξ2 + ξ3ξ1 + ξ1 in degree 9 Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 14 / 21 Summary of Work • Let Dn be the dimension of A∗ in degree n χ • Crossley and Whitehouse bounded the dimension dn of A∗ in degree n as Bound on dn D D n d D n−1 2 ≤ n ≤ n − 2 • Understanding Dn gives strong bounds on dn Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 15 / 21 Summary of Work Dimension of degree n part of dual Steenrod algebra 106 1:2 ⋅ 0:9 Dn 0:6 0:3 0 0 100 200 300 400 500 n • Dn grows faster than polynomial, but still infra-exponential Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 16 / 21 Asymptotic Behavior ln2 n D exp n ∼ 2 ln 2 Summary of Work • Similar Diophantines suggest the following asymptotic behavior: Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 17 / 21 Summary of Work • Similar Diophantines suggest the following asymptotic behavior: Asymptotic Behavior ln2 n D exp n ∼ 2 ln 2 Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 17 / 21 χ • Full description of A∗ Future Work • Asymptotics for rest of cohomology groups Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 18 / 21 Future Work • Asymptotics for rest of cohomology groups χ • Full description of A∗ Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 18 / 21 Acknowledgements I would like to sincerely thank the following: • Sanath Devalapurkar • Dr.
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