Question

Calculate the homology groups of Tⁿ=xⁿS and Sⁿ *Sᵐ, for ( n, m ). using CW complex and cellular homology.

Answer 1

To calculate the homology groups of Tⁿ=xⁿS and Sⁿ *Sᵐ, we can use CW complex and cellular homology. Construct a CW complex for each given space and calculate the homology groups by applying the cellular homology algorithm.

Step-by-step explanation:

The homology groups of Tⁿ=xⁿS and Sⁿ *Sḿ are calculated using CW complex and cellular homology. To calculate the homology groups, we first construct a CW complex for the given spaces and then use cellular homology to determine the homology groups.

Here are the steps to calculate the homology groups:

1. For Tⁿ=xⁿS, we can construct a CW complex with one 0-cell, one 1-cell for each coordinate axis, one 2-cell for each face of the n-dimensional cube, and so on up to n-cells. The homology groups can be found by computing the cellular homology using the boundary maps.
2. For Sⁿ *Sḿ, we can construct a CW complex with one 0-cell, one 1-cell for each coordinate axis of Sⁿ, one 2-cell for each face of the n-dimensional cube, and so on up to n-cells. Additionally, we can add m-dimensional cells for each coordinate axis of Sḿ. Again, the homology groups can be found by computing the cellular homology using the boundary maps.

By applying the cellular homology algorithm to the constructed CW complexes, we can calculate the homology groups of Tⁿ=xⁿS and Sⁿ *Sḿ.