Continuous Maps, Homomorphisms and Cyclic Groups
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Let f: S^n --> S^n be a continuous map.
Consider the induced homomorphism f*: H~_n (S^n) --> H~_n (S^n), where
H~_n is a reduced homology group. Then from the fact that H~_n (S^n) is an infinite
cyclic group, it follows that there is a unique integer d such that
f*(u) = du for any u in H~_n (S^n).
How exactly does "there is a unique integer d such that..." follow from
"H~_n (S^n) is an infinite cyclic group"?
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Solution Summary
Continuous Maps, Homomorphisms and Cyclic Groups are investigated.
Solution Preview
Call H~_n(S^n) G for short.
Take a generator g of the group G, so G = {1, g, g^2, g^3 ,...., g^{-1}, g^{-2},... ...
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