Persistent cohomology operations and Gromov-Hausdorff estimates

Authors

Anibal M. Medina-Mardones and Ling Zhou

Abstract

We establish the foundations of the theory of persistent cohomology operations, derive decomposition formulas for wedge sums and products, and prove their Gromov-Hausdorff stability. We use these results to construct pairs of Riemannian pseudomanifolds for which the Gromov-Hausdorff estimates derived from persistent cohomology operations are strictly sharper than those obtained using persistent homology.

Bibliography: technical aspects

This project uses the package biblatex to handle the bibliography, therefore it needs biber as the bibliography tool, instead of bibtex. Additionally, it is suggested that all bib entries include a DOI and/or URL.

Best LaTeX practices

1. Use \colon instead of : for defining functions.

2. Use \mid for sets or group presentations. For example, \set{a \in A \mid a=b} is superior to \set{a \in A | a=b}.

3. Leave space around most operators. For example, f \colon X \to Y is superior to f:X\to Y. Possible exceptions are guided by readability, for example:

   \begin{equation}
       \sum_{i=1}^n = \frac{n(n+1)}{2}.
   \end{equation}

4. No punctuation inside math mode. Use $a+b$, instead of $a+b,$.

5. No line breaks in the middle of sentences.

6. Use line breaks at the end of each sentences.

7. No double blank spaces or trailing blank spaces.

8. Use tab instead of blank spaces for indentation.

9. All named environments with the possible exception of document should be tabbed. For example,

   \begin{lemma*}
       Statement.
   \end{lemma*}

10. Do not use $$...$$ for display equations; instead, use one of the environments in the AMS display equation family or

    \[
    \]

11. Leave a blank line before and after most environments. Possible exceptions are AMS display equation family.

12. Leave a blank line before and after sections and subsections.

13. Unless a there is a good reason not to, use \dots instead of either \ldots or \cdots or others. The amsmath package uses nearby operators to choose the right type of dots for \dots.