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 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 • Œ ‘ + Œ ‘ = Œ ‘ a21 a22 b21 b22 a21 + b21 a22 + b22

a11 a12 b11 b12 a11b11 + a12b12 a11b12 + a12b22 • Œ ‘ Œ ‘ = Œ ‘ 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 ) • 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 (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. Tanya Khovanova • MIT PRIMES-USA Program • MIT Math Department • My family

Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 19 / 21 References

Cassels, J.W.S. and Frohlich, A. Algebraic Number Theory. 1967 Crossley, M.D. and Whitehouse, S. On conjugation invariants in the dual Steenrod algebra Proceedings of the American Mathematical Society, vol. 128, 2000, pp. 2809-2818. Milnor, John The Steenrod algebra and its dual Ann. Math., , 67, (1958), 150-171. MR 20:6092 2809-2818. [Image] Wikimedia Commons Mug and Torus morph

Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 20 / 21 Thank you! Questions?

Ryan Kim (PRIMES) Cohomology of Dual Steenrod Algebra May 19, 2018 21 / 21