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, On minimal round functions, Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.

Title	t-cat
Canonical name	Tcat
Date of creation	2013-03-22 15:54:54
Last modified on	2013-03-22 15:54:54
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	7
Author	juanman (12619)
Entry type	Definition
Classification	msc 55M30
Related topic	RoundComplexity