# Steenrod Squares over an Infinite Projective Space

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In summary, an infinite projective space is a geometric space denoted by P^n in which each point represents a line in n-dimensional space. Steenrod squares are operations used in algebraic topology to study the cohomology of topological spaces, and they can be applied to calculate the cohomology of infinite projective spaces. Studying Steenrod squares over an infinite projective space is significant as it allows for a deeper understanding of these spaces and has important applications in mathematics. Ongoing research in this area includes generalizing Steenrod squares to other types of spaces and exploring connections with other areas of mathematics.
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Let ##u## be a generator of ##H^1(\mathbb{R} P^\infty; \mathbb{F}_2)##. Prove the relations $$\text{Sq}^i(u^n) =\binom{n}{i} u^{n+i}$$

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We can induct on ##n.## The base case is clear. Next, assuming the formula to be true for exponent ##n## (and for all ##i##), we have:

$$\text{Sq}^i(u^{n+1})=\text{Sq}^i(u^n\cup u)=\sum_{a+b=i} Sq^a(u^n) \cup Sq^b(u).$$

Since ##Sq^b(u)## vanishes when ##b>1##, the only terms in the sum are ##\text{Sq}^i(u^n)\cup \text{Sq}^0(u)+\text{Sq}^{i-1}(u^n) \text{Sq}^1(u)=\binom{n}{i}u^{n+i+1}+\binom{n}{i-1}u^{n+i+1}.## So we just need to verify that ##\binom{n}{i}+\binom{n}{i-1}=\binom{n+1}{i}## (in fact we only need to check that it is true mod 2, but it is true over the integers). The number of ways of picking ##i## items from ##n+1## items is the number of ways of picking ##i## items where the first item is included (## \binom{n}{i-1} ## ways) plus the number of ways of picking ##i## items from ##n+1## where the first item is not picked (## \binom{n}{i}## ways).

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## 1. What are Steenrod Squares?

Steenrod Squares are a mathematical concept used in algebraic topology to study cohomology groups. They were introduced by mathematician Norman Steenrod in the 1940s.

## 2. What is an Infinite Projective Space?

Infinite Projective Space is a mathematical concept used in algebraic geometry and topology to study spaces that are infinite in size. It is denoted by the symbol ∞Pn and can be thought of as a generalization of ordinary Euclidean space.

## 3. How are Steenrod Squares used in Infinite Projective Space?

Steenrod Squares are used in Infinite Projective Space to study the cohomology groups of these spaces. They allow for a better understanding of the topology and geometry of these spaces, and can be used to prove important theorems and results.

## 4. What are the properties of Steenrod Squares over Infinite Projective Space?

Steenrod Squares over Infinite Projective Space have many important properties, including the Cartan formula, the Adem relations, and the Milnor-Moore theorem. These properties allow for the calculation and manipulation of these squares in various contexts.

## 5. Are there any applications of Steenrod Squares over Infinite Projective Space?

Yes, Steenrod Squares have many applications in mathematics and other fields. They are used in algebraic topology, algebraic geometry, and theoretical physics, among others. They also have connections to other areas of mathematics, such as homotopy theory and Lie groups.

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