ABSTRACT
Let x = s 3 \ N ° ( K ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq396.tif"/> be the complement in S3 of an open tubular neighborhood of the figure eight knot K. Denote by Xn the n-fold cyclic cover of X, and by Σn the n-fold branched cyclic cover of S3 branched over K (cf., [4]). In this paper we consider the question of whether Σn has a finite cover with positive first Betti number. We prove:
For n ≥ 5, Σn has a finite cover ∑ ˜ n → ∑ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq397.tif"/> with rank H 1 ( ∑ ˜ n ) > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq398.tif"/> .
Since Σ1 = S3 and π 1 ( ∑ 2 ) ≅ ℤ / 5 ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq399.tif"/> , the problem is posed for n ≥ 3. Hempel [2] has proved this result for n odd, using different methods.
