[107] On Invariants of Manifolds

  • Raoul Bott
  • Loring W. Tu
Part of the Contemporary Mathematicians book series (CM)


It is a great pleasure to participate in these festive proceedings. Of course from my much advanced vantage point they resemble nothing more than a Bar Mitzvah- a rite of passage of the young. In short, they signify that finally we can welcome you, Joe, and you, Bob, into the community of - shall we say - mature mathematicians. That is as far as I am willing to go, for I doubt that true maturity can ever be attained by mathematicians.

Returning to Princeton after forty years, it is difficult to refrain from reminiscing - especially when one meets such wonderful old comrades in arms as Don Spencer upon entering the hall, and I have therefore chosen to speak about manifold invariants in the context of the forty years that have passed since I first encountered them here.

When our forefathers spoke of an invariant they usually meant a number, and so every discussion of topological invariants must really start with the Euler number of a cell complex:

$$\chi \left(...


  1. (1) Axelroad & Singer, Chern-Simons perturbation theory, MIT Preprint, October 1991.Google Scholar
  2. (2) Dror Bar-Natan, Perturbative aspects of Chern-Simons topological quantum field theory. Ph.D. Thesis. Princeton Univ., June 1991. Dept. of Math.Google Scholar
  3. (3) Dror Bar-Natan, On Vasiliev Knot Invariants, Harvard Preprint, Oct. 16, 1992.Google Scholar

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Authors and Affiliations

  • Raoul Bott
  • Loring W. Tu
    • 1
  1. 1.Department of MathematicsTufts UniversityMedfordUSA

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