# t-cat

The t-cat of a topological space $X$ is the minimal number of open sets that cover $X$ such that each open set in the cover has the homotopy type of the unit circle $S^{1}$. This means that for each open set $U$, the inclusion $U\lx@stackrel{{\scriptstyle i}}{{\hookrightarrow}}X$ is homotopic to some factorization $U\lx@stackrel{{\scriptstyle a}}{{\to}}S^{1}\lx@stackrel{{\scriptstyle b}}{{\to% }}X$, i.e.

 $i\simeq b\circ a.$

When $X$ is manifold, this is related to the round complexity of $X$.

## References

• 1 D. Siersma, G. Khimshiasvili, , Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.
Title t-cat Tcat 2013-03-22 15:54:54 2013-03-22 15:54:54 juanman (12619) juanman (12619) 7 juanman (12619) Definition msc 55M30 RoundComplexity