Topological structures on products of spheres (Ex)

Exercise 0.1.

1. Determine $\mathcal{S}^{TOP}(S^p \times S^q)$$<wikitex>; {{beginthm|Exercise}} # Determine $\mathcal{S}^{TOP}(S^p \times S^q)$ for $p, q$ both greater than \mathcal{S}^{TOP}(S^p \times S^q)$ for $p, q$$p, q$ both greater than $1$$1$ and $p + q > 4$$p + q > 4$.
2. When $(p, q) = (4k, 4k)$$(p, q) = (4k, 4k)$ determine the action of the group of homotopy self-equivalences, $\textup{hAut}(S^{4k} \times S^{4k})$$\textup{hAut}(S^{4k} \times S^{4k})$, on $\mathcal{S}^{TOP}(S^{4k} \times S^{4k})$$\mathcal{S}^{TOP}(S^{4k} \times S^{4k})$.

Hint 0.2. For the second part, you may wish to look at Normal maps and submanifolds (Ex).

The first part may be found in [Ranicki2009a, Ex 3.6], [Kreck&Lück2009, Section 7] and [Crowley2010].

Exercise 0.3. Repeat the above exercise for as many different values of $(p, q)$$(p, q)$ as you can:

1. $(p, q) = (2j+1, 4k)$$(p, q) = (2j+1, 4k)$
2. $(p, q) = (4j, 4k)$$(p, q) = (4j, 4k)$
3. $(p, q) = (i, 4k+2)$$(p, q) = (i, 4k+2)$ as $i$$i$ ranges over $i = 2j +1, i = 4j$$i = 2j +1, i = 4j$ and $i = 4j+2$$i = 4j+2$

$ and $p + q > 4$. # When $(p, q) = (4k, 4k)$ determine the action of the group of homotopy self-equivalences, $\textup{hAut}(S^{4k} \times S^{4k})$, on $\mathcal{S}^{TOP}(S^{4k} \times S^{4k})$. {{endthm}} {{beginrem|Hint}} For the second part, you may wish to look at [[Normal maps and submanifolds (Ex)]]. The first part may be found in {{citeD|Ranicki2009a|Ex 3.6}}, {{citeD|Kreck&Lück2009|Section 7}} and {{citeD|Crowley2010}}. {{endrem}} {{beginthm|Exercise}} Repeat the above exercise for as many different values of $(p, q)$ as you can: # $(p, q) = (2j+1, 4k)$ # $(p, q) = (4j, 4k)$ # $(p, q) = (i, 4k+2)$ as $i$ ranges over $i = 2j +1, i = 4j$ and $i = 4j+2$ {{endthm}} [[Category:Exercises]] [[Category:Exercises without solution]]\mathcal{S}^{TOP}(S^p \times S^q) for $p, q$$p, q$ both greater than $1$$1$ and $p + q > 4$$p + q > 4$.

When $(p, q) = (4k, 4k)$$(p, q) = (4k, 4k)$ determine the action of the group of homotopy self-equivalences, $\textup{hAut}(S^{4k} \times S^{4k})$$\textup{hAut}(S^{4k} \times S^{4k})$, on $\mathcal{S}^{TOP}(S^{4k} \times S^{4k})$$\mathcal{S}^{TOP}(S^{4k} \times S^{4k})$.