A question. According to this lecture, the points of the Möbius band as a moduli space correspond to the lines of the affine plane. Such a line is defined by ax + by + c = 0, and, considering the homogeneous coefficients [a : b : c], it seems to me that this is equivalent to the punctured projective plane, because [a : b : c] = [0 : 0 : 1] must be excluded, since it does not define an affine line. Is this correct ? And if yes, the projective plane also consisting of a Möbius band of a disk glued together, is it true that the closed disk can be, in this case, topologically collapsed to a single point ? Thank you in advance.
Hi Loic, Yes that is right I believe: the projective plane can be viewed as a Mobius band with a disk glued to the boundary. And any closed disk can be topologically collapsed to a single point.
@@njwildberger Many thanks for answering me! I though a bit more about it, and I think now that the distinction has to be made about removing a closed or an open disk from the projective plane. Removing an open disk yields the Möbius band (which has a boundary), whereas removing a closed disk (or a point, which is also a closed set) yields a boundary-less Möbius band, or, equivalently, a punctured projective plane.
wonderful lesson, thank you! One question: In your identification P^1 with S^1 why didn't you identify antipodal points of S^1 in the same way you do in one higher dimension?
what a breath of fresh air. Thank you!
A question. According to this lecture, the points of the Möbius band as a moduli space correspond to the lines of the affine plane. Such a line is defined by ax + by + c = 0, and, considering the homogeneous coefficients [a : b : c], it seems to me that this is equivalent to the punctured projective plane, because [a : b : c] = [0 : 0 : 1] must be excluded, since it does not define an affine line. Is this correct ? And if yes, the projective plane also consisting of a Möbius band of a disk glued together, is it true that the closed disk can be, in this case, topologically collapsed to a single point ? Thank you in advance.
Hi Loic, Yes that is right I believe: the projective plane can be viewed as a Mobius band with a disk glued to the boundary. And any closed disk can be topologically collapsed to a single point.
@@njwildberger Many thanks for answering me! I though a bit more about it, and I think now that the distinction has to be made about removing a closed or an open disk from the projective plane. Removing an open disk yields the Möbius band (which has a boundary), whereas removing a closed disk (or a point, which is also a closed set) yields a boundary-less Möbius band, or, equivalently, a punctured projective plane.
wonderful lesson, thank you! One question: In your identification P^1 with S^1 why didn't you identify antipodal points of S^1 in the same way you do in one higher dimension?
S^1 with antipodal points identified is still S^1; not true in one higher dimension.
thanks for share, easy explained. Love it.