Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider a continuous map $f: X \rightarrow Y$, where $X$ and $Y$ are compact $CW$ complexes. Are there interesting examples of $f$ such that the pre-image of a cell in $X$ is not a cell in $Y$?

share|improve this question
    
What's an interesting example? –  Qiaochu Yuan Feb 16 at 8:44
    
Any map can satisfy this if you are willing the change the CW structure of $X$. Just take the cells and start subdividing, then the preimages won't be cells anymore. For example, take $X = S^1$ with two $0$-cells and two $1$-cells, then take $Y = S^1$ with one $0$-cell and one $1$-cell, and consider the identity map $S^1 \to S^1$. –  Justin Young Mar 13 at 14:01

1 Answer 1

Why not? Take any continuous map $f:X\to Y$ and if it doesn't already have this property, perturb it slightly. For instance, if we realize $\mathbb{S}^1$ with one zero-cell at $1$ and one one-cell connecting $1$ to itself, any continuous self-map which does not take $1$ to itself meets your criteria.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.