How does homotopy groups of wedge sums relate to the individual spheres?

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    2015
In summary, homotopy groups are algebraic structures that measure connectivity in topological spaces and are related to wedge sums. The homotopy groups of wedge sums can be expressed as direct sums of the homotopy groups of spheres, making spheres building blocks for understanding them. The homotopy groups of wedge sums can be computed using the homotopy groups of individual spheres, and the fundamental group of a wedge sum is isomorphic to the direct sum of the fundamental groups of the individual spaces. In special cases where the individual spaces are spheres of the same dimension, the homotopy groups of the wedge sum are isomorphic to the homotopy groups of the individual spheres. This is known as the wedge sum of homotopy
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Euge
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Thanks to all MHB members who have participated this year in the Graduate POTWs. (Happy)
Here is the last problem of the year, concerning homotopy groups:

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Explain why $\pi_r(S^m \lor S^n) \approx \pi_r(S^m) \oplus \pi_r(S^n)$ for $2 \le r < m + n - 1$.

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No one answered this week's problem. You can read my solution below.

For $r \ge 2$, there is an isomorphism $\pi_r(S^m\lor S^n) \approx \pi_r(S^m) \oplus \pi_r(S^n) \oplus \pi_r(S^m\times S^n, S^m\lor S^n)$. The product $S^m \times S^n$ has an $(m + n)$-dimensional CW-complex structure with $1$ $0$-cell, $1$ $m$-cell, $1$ $n$-cell, and $1$ $(m+n)$-cell. The $m + n - 1$ skeleton is $S^m \lor S^n$, and hence $\pi_r(S^m \times S^n, S^m\lor S^n) = 0$ for $r < m + n - 1$. Therefore, $\pi_r(S^m \lor S^n) \approx \pi_r(S^m) \lor \pi_r(S^n)$ for $2\le r < m + n - 1$.
 

FAQ: How does homotopy groups of wedge sums relate to the individual spheres?

1. What are homotopy groups and how are they related to wedge sums?

Homotopy groups are algebraic structures that measure the connectivity of a topological space. They are related to wedge sums because the homotopy groups of a wedge sum are determined by the homotopy groups of the individual spaces that are being wedged together.

2. How do the homotopy groups of wedge sums relate to the homotopy groups of spheres?

The homotopy groups of wedge sums can be expressed as direct sums of the homotopy groups of spheres. This means that the homotopy groups of spheres are building blocks for understanding the homotopy groups of wedge sums.

3. Can the homotopy groups of wedge sums be computed using the homotopy groups of individual spheres?

Yes, the homotopy groups of wedge sums can be computed using the homotopy groups of individual spheres. This is because the homotopy groups of wedge sums are determined by the homotopy groups of the individual spaces that are being wedged together.

4. How do the homotopy groups of wedge sums relate to the fundamental group of the wedge sum?

The fundamental group of a wedge sum is isomorphic to the direct sum of the fundamental groups of the individual spaces being wedged together. This means that the fundamental group of a wedge sum can be understood in terms of the fundamental groups of the individual spheres.

5. Are there any special cases where the homotopy groups of wedge sums can be simplified?

Yes, if the individual spaces being wedged together are spheres of the same dimension, then the homotopy groups of the wedge sum are isomorphic to the homotopy groups of the individual spheres. This simplification is known as the wedge sum of homotopy equivalent spaces.

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