Suppose $X$ is a contractible space and $A \subset X$ with $A$ being homotopy equivalent to $S^2$. Prove that there is no retraction $X \rightarrow A$

score 46 · Answer 12 · 2026-04-16 20:05:49

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For each $\sigma \in \mathrm{S}_{m}$, $p \mapsto \sigma \cdot p$ is an automorphism of $R[X_{1}, \ldots, X_{m}]$

Published on 16 Apr 2026 - 20:05

score 43 · Answer 13 · 2026-05-17 04:35:14

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An elaboration of Thm. 8.11(c) proof.

Published on 17 May 2026 - 4:35

score 33 · Answer 14 · 2026-04-17 07:02:58

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Linearisation of $x'''(t)+2x'(t)+x^2(t)=0$ about the initial point $\underline{x}(0)=0$

Published on 17 Apr 2026 - 7:02

score 35 · Answer 15 · 2026-04-17 11:52:40

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Proof-Verification: if $f''_{xy}(x,y),f''_{yx}(x,y)$ is continuous at $(x_0,y_0)$，then $f''_{xy}(x_0,y_0)=f''_{yx}(x_0,y_0).$

Published on 17 Apr 2026 - 11:52

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Suppose $X$ is a contractible space and $A \subset X$ with $A$ being homotopy equivalent to $S^2$. Prove that there is no retraction $X \rightarrow A$

For each $\sigma \in \mathrm{S}_{m}$, $p \mapsto \sigma \cdot p$ is an automorphism of $R[X_{1}, \ldots, X_{m}]$

An elaboration of Thm. 8.11(c) proof.

Linearisation of $x'''(t)+2x'(t)+x^2(t)=0$ about the initial point $\underline{x}(0)=0$

Proof-Verification: if $f''_{xy}(x,y),f''_{yx}(x,y)$ is continuous at $(x_0,y_0)$，then $f''_{xy}(x_0,y_0)=f''_{yx}(x_0,y_0).$

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