Relative Topological Complexity and Configuration Spaces

Abstract: Given a space

X

, the topological complexity of

X

, denoted by

T C (X)

, can be viewed as the minimum number of "continuous rules" needed to describe how to move between any two points in

X

. Given subspaces

Y_{1}

and

Y_{2}

of

X

, there is a "relative" version of topological complexity, denoted by

T C_{X} (Y_{1} i m e s Y_{2})

, in which one only considers paths starting at a point

y_{1} i n Y_{1}

and ending at a point

y_{2} i n Y_{2}

, but the path from

y_{1}

to

y_{2}

can pass through any point in

X

. We discuss general results that provide relative analogues of well-known results concerning

T C (X)

before focusing on the case in which we have

Y_{1} = Y_{2} = C^{n} (Y)

, the configuration space of

n

points in some space

Y

, and

X = C^{n} (Y i m e s I)

, the configuration space of

n

points in

Y i m e s I

, where

I

denotes the interval

[0, 1]

. Our main result shows

T C_{C^{n} (Y i m e s I)} (C^{n} (Y) i m e s C^{n} (Y))

is bounded above by

T C (Y^{n})

and under certain hypotheses is bounded below by

T C (Y)

.

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