Geodesic
Analytic Evaluation of Certain Characteristic Classes(1)
Canadian mathematical bulletin, Tome 16 (1973) no. 2, pp. 219-223

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We have an alternative proof of the following result of Kervaire [2]: Let V→M be a real vector bundle with fibre dimension n≥4k+l over a compact 4k-manifold. Suppose V restricted to M — {x} is trivial. Choose a Riemann structure for V and an orthonormal frame for V restricted to M —{x}. Thus the obstruction to extending the frame smoothly over M is an element λ in π4k+1SO(n))≅Z. Then up to sign the evaluation of the kth Pontrayagin class Pk on M is ak(2k— 1)!. λ, where ak is 1 or 2 depending upon whether k is even or odd. 
Jeffries, Clark D. Analytic Evaluation of Certain Characteristic Classes(1). Canadian mathematical bulletin, Tome 16 (1973) no. 2, pp. 219-223. doi: 10.4153/CMB-1973-038-5
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     title = {Analytic {Evaluation} of {Certain} {Characteristic} {Classes(1)}},
     journal = {Canadian mathematical bulletin},
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     year = {1973},
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     number = {2},
     doi = {10.4153/CMB-1973-038-5},
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