Deformation retraction

cjerdonek authored on Nov. 4, 2020, 5:23 p.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 the difference between a deformation retraction that is strong and one that is not strong. Both the left and the right show a deformation retraction from a square disk to a point on the disk's boundary on the lower left. However, the retraction on the left is strong while the one on the right is not. 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 is not strong.	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.

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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 the difference between a deformation retraction that is strong and one that is not strong. Both the left and the right show a deformation retraction from a square disk to a point on the disk's boundary on the lower left. However, the retraction on the left is strong while the one on the right is not.

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 is not strong.

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.

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

cjerdonek / Deformation retraction bce664dff12115d9f40bcf429d48c9231caf6552