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Ali Caglayan
Ali Caglayan
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Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I always thought that $S^0$ was the odd one out due to it not being connected however after reading page 4 of "H-spaces from a homotopy point of view" I see that it is claimed $0$-connectedness can be weakened to $\pi_0(X)$ being a group.

Does this mean the real Hopf fibration can be constructed using the Hopf construction?

If so could I have some modern references?

This question originally stemmed from the discussion here on the nforum.

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I always thought that $S^0$ was the odd one out due to it not being connected however after reading page 4 of "H-spaces from a homotopy point of view" I see that it is claimed $0$-connectedness can be weakened to $\pi_0(X)$ being a group.

Does this mean the real Hopf fibration can be constructed using the Hopf construction?

If so could I have some modern references?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I always thought that $S^0$ was the odd one out due to it not being connected however after reading page 4 of "H-spaces from a homotopy point of view" I see that it is claimed $0$-connectedness can be weakened to $\pi_0(X)$ being a group.

Does this mean the real Hopf fibration can be constructed using the Hopf construction?

If so could I have some modern references?

This question originally stemmed from the discussion here on the nforum.

capitalized Hopf
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YCor
YCor
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Does the hopfHopf construction work for $S^0$?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the hopfHopf fibrations for $S^1$, $S^3$ and $S^7$. I always thought that $S^0$ was the odd one out due to it not being connected however after reading page 4 of "H-spaces from a homotopy point of view" I see that it is claimed $0$-connectedness can be weakened to $\pi_0(X)$ being a group.

Does this mean the real hopfHopf fibration can be constructed using the hopfHopf construction?

If so could I have some modern references?

Does the hopf construction work for $S^0$?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the hopf fibrations for $S^1$, $S^3$ and $S^7$. I always thought that $S^0$ was the odd one out due to it not being connected however after reading page 4 of "H-spaces from a homotopy point of view" I see that it is claimed $0$-connectedness can be weakened to $\pi_0(X)$ being a group.

Does this mean the real hopf fibration can be constructed using the hopf construction?

If so could I have some modern references?

Does the Hopf construction work for $S^0$?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I always thought that $S^0$ was the odd one out due to it not being connected however after reading page 4 of "H-spaces from a homotopy point of view" I see that it is claimed $0$-connectedness can be weakened to $\pi_0(X)$ being a group.

Does this mean the real Hopf fibration can be constructed using the Hopf construction?

If so could I have some modern references?

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Ali Caglayan
Ali Caglayan
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Ali Caglayan
Ali Caglayan
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