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Fixed typo: the $n$-simplex has $n+1$ vertices.
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Allen Hatcher
Allen Hatcher
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The labeling gives a piecewise linear map from $S^n$ to a simplex $\Delta$ with $n$$n+1$ vertices, where label $k$ means the point is sent to vertex $v_k$, and we extend linearly on simplices. The map must have degree $0$ because $\Delta$ is contractible. The degree can be computed from the preimage of any point in the interior of $\Delta$, and it is the difference between the number of positive simplices and the number of negative simplices.

The labeling gives a piecewise linear map from $S^n$ to a simplex $\Delta$ with $n$ vertices, where label $k$ means the point is sent to vertex $v_k$, and we extend linearly on simplices. The map must have degree $0$ because $\Delta$ is contractible. The degree can be computed from the preimage of any point in the interior of $\Delta$, and it is the difference between the number of positive simplices and the number of negative simplices.

The labeling gives a piecewise linear map from $S^n$ to a simplex $\Delta$ with $n+1$ vertices, where label $k$ means the point is sent to vertex $v_k$, and we extend linearly on simplices. The map must have degree $0$ because $\Delta$ is contractible. The degree can be computed from the preimage of any point in the interior of $\Delta$, and it is the difference between the number of positive simplices and the number of negative simplices.

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Douglas Zare
Douglas Zare
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The labeling gives a piecewise linear map from $S^n$ to a simplex $\Delta$ with $n$ vertices, where label $k$ means the point is sent to vertex $v_k$, and we extend linearly on simplices. The map must have degree $0$ because $\Delta$ is contractible. The degree can be computed from the preimage of any point in the interior of $\Delta$, and it is the difference between the number of positive simplices and the number of negative simplices.