DGA map

DGA map

Consider the map $f\colon S^{2n}\times S^{2n}\to S^{4n}$ obtained by collapsing $n$-skeleton of $S^{2n}\times S^{2n}$ to a point. Its Sullivan representative can be defined as follows.

First we define Sullivan models of $S^{4n}$ and $S^{2n}\times S^{2n}$. They can be taken as $(\wedge(x,y), d)$ with $dx=0$ and $dy=x^2$, and $(\wedge(a_1, b_1, a_2, b_2), d)$ with $da_i=0$ and $db_i=a_i^2$, respectively.

DGAlgebraMap.kt

val n = 1
val matrixSpace = SparseMatrixSpaceOverRational

// define a Sullivan model of the 4n-sphere
val sphereIndeterminateList = listOf(
    Indeterminate("x", 4 * n),
    Indeterminate("y", 8 * n - 1),
)
val sphere = FreeDGAlgebra.fromMap(matrixSpace, sphereIndeterminateList) { (x, y) ->
    mapOf(y to x.pow(2))
}

// define a Sullivan model of the product of two 2n-spheres
val sphereProductIndeterminateList = listOf(
    Indeterminate("a1", 2 * n),
    Indeterminate("b1", 4 * n - 1),
    Indeterminate("a2", 2 * n),
    Indeterminate("b2", 4 * n - 1),
)
val sphereProduct = FreeDGAlgebra.fromMap(matrixSpace, sphereProductIndeterminateList) { (a1, b1, a2, b2) ->
    mapOf(b1 to a1.pow(2), b2 to a2.pow(2))
}

Sullivan representative of the map $f$ is a DGA map $\varphi\colon\wedge(x,y)\to\wedge(a_1, b_1, a_2, b_2)$ given by $\varphi(x)=a_1a_2$ and $\varphi(y)=a_1^2b_2$. In kohomology, this can be defined and used as follows:

DGAlgebraMap.kt

val (x, y) = sphere.generatorList
val (a1, b1, a2, b2) = sphereProduct.generatorList
val valueList = sphereProduct.context.run {
    listOf(a1 * a2, a1.pow(2) * b2)
}
val f = sphere.getDGAlgebraMap(sphereProduct, valueList)
sphere.context.run {
    // This 'context' is necessary for pow(2) and cohomologyClass()
    println(f(x)) // a1a2
    println(f(x.pow(2))) // a1^2a2^2
    println(f.inducedMapOnCohomology(x.cohomologyClass())) // [a1a2]
    println(f.inducedMapOnCohomology(x.pow(2).cohomologyClass())) // 0
}