Tucker's lemma

In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker.

Let T be a triangulation of the closed n-dimensional ball

B

n

{\displaystyle B_{n}}

. Assume T is antipodally symmetric on the boundary sphere

S

n

−

1

{\displaystyle S_{n-1}}

. That means that the subset of simplices of T which are in

S

n

−

1

{\displaystyle S_{n-1}}

provides a triangulation of

S

n

−

1

{\displaystyle S_{n-1}}

where if σ is a simplex then so is −σ.

Let

L

:

V

(

T

)

→

{

+

1

,

−

1

,

+

2

,

−

2

,

.

.

.

,

+

n

,

−

n

}

{\displaystyle L:V(T)\to \{+1,-1,+2,-2,...,+n,-n\}}

be a labeling of the vertices of T which is an odd function on

S

n

−

1

{\displaystyle S_{n-1}}

, i.e.,

L

(

−

v

)

=

−

L

(

v

)

{\displaystyle L(-v)=-L(v)}

for every vertex

v

∈

S

n

−

1

{\displaystyle v\in S_{n-1}}

.

Then Tucker's lemma states that T contains a complementary edge - an edge (a 1-simplex) whose vertices are labelled by the same number but with opposite signs.

Proofs

The first proofs were non-constructive, by way of contradiction.