- #1

Ben2

- 37

- 9

- Homework Statement
- From Hatcher, "Algebraic Topology" p. 3 line 32: "...[T]hree graphs [two circles connected with a line segment, a horizontal figure 8 and an ellipse bisected by a vertical line segment] are all homotopy equivalent since they are deformation retracts of the same space..."

- Relevant Equations
- ##F(x,t) = f_t(x)##, fg\cong\\mathbb{1}##, ##X\congY##, where the author's "congruence" sign has an additional line segment

For F: X x I-->Y, defined by F(x,t) = y, next define G: Y x I-->X by G(y,u) = x. Then for t = u, we have

F[G(y,t),t] = F{G[F(x,t),t]}, which will ideally be ##\mathbb{1}##. Given Hatcher's definitions pp. 2-3, to me it's not clear how to "invert" a homotopy without an inverse function--let alone how to "invert" a deformation retract. The latter seems to be a set of continuous projection maps. Thanks again for all feedback!

F[G(y,t),t] = F{G[F(x,t),t]}, which will ideally be ##\mathbb{1}##. Given Hatcher's definitions pp. 2-3, to me it's not clear how to "invert" a homotopy without an inverse function--let alone how to "invert" a deformation retract. The latter seems to be a set of continuous projection maps. Thanks again for all feedback!