Deformation retraction

cjerdonek authored on Nov. 26, 2020, 3:50 a.m.

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In mathematics (and specifically topology), a deformation retraction captures the idea of deforming a space over time into a subspace of itself. This figure shows what it means for a deformation retraction to be strong versus not strong. Both the left and right show a deformation retraction from a square disk to the subspace defined by its bottom edge. The retraction on the left is strong. However, the one on the right isn't strong because the edge subspace doesn't map to itself throughout the entire deformation. Specifically, given a topological space \(X\) and a subspace \(A\) of \(X\), a deformation retraction is a homotopy from the identity map on \(X\) to a map from \(X\) to \(A\) that is the identity on \(A\). In addition, if the homotopy is the identity on \(A\) for all maps in the family (so not just the first and final maps), then the deformation retraction is said to be a strong deformation retraction. The figure above illustrates the difference between a deformation retraction that is strong and one that isn't.	In mathematics (and specifically topology), a deformation retraction captures the idea of deforming a space over time into a subspace of itself. Specifically, given a topological space \(X\) and a subspace \(A\) of \(X\), a deformation retraction is a homotopy from the identity map on \(X\) to a map from \(X\) to \(A\) that is the identity on \(A\). In addition, if the homotopy is the identity on \(A\) for all maps in the family (so not just the first and final maps), then the deformation retraction is said to be a strong deformation retraction. This figure shows what it means for a deformation retraction to be strong versus not strong. Both the left and right show a deformation retraction from a square disk to the subspace defined by its bottom edge. The retraction on the left is strong. However, the one on the right isn't strong because the edge subspace doesn't map to itself throughout the entire deformation. The figure above illustrates the difference between a deformation retraction that is strong and one that isn't.

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In mathematics (and specifically topology), a deformation retraction captures the idea of deforming a space over time into a subspace of itself.

This figure shows what it means for a deformation retraction to be strong versus not strong. Both the left and right show a deformation retraction from a square disk to the subspace defined by its bottom edge. The retraction on the left is strong. However, the one on the right isn't strong because the edge subspace doesn't map to itself throughout the entire deformation.

Specifically, given a topological space \(X\) and a subspace \(A\) of \(X\), a deformation retraction is a homotopy from the identity map on \(X\) to a map from \(X\) to \(A\) that is the identity on \(A\). In addition, if the homotopy is the identity on \(A\) for all maps in the family (so not just the first and final maps), then the deformation retraction is said to be a strong deformation retraction.

The figure above illustrates the difference between a deformation retraction that is strong and one that isn't.

In mathematics (and specifically topology), a deformation retraction captures the idea of deforming a space over time into a subspace of itself.

The figure above illustrates the difference between a deformation retraction that is strong and one that isn't.

cjerdonek / Deformation retraction 938cb49ccaf6133f20b08c193786bff849bb76ae