## Algebraic & Geometric Topology

### Index theory of the de Rham complex on manifolds with periodic ends

#### Abstract

We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover $X̃→X$. The completion of this complex in exponentially weighted $L2$ norms is Fredholm for all but finitely many exceptional weights determined by the eigenvalues of the covering translation map $H∗(X̃)→H∗(X̃)$. We calculate the index of this weighted de Rham complex for all weights away from the exceptional ones.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 6 (2014), 3689-3700.

Dates
Revised: 31 July 2014
Accepted: 26 August 2014
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513716053

Digital Object Identifier
doi:10.2140/agt.2014.14.3689

Mathematical Reviews number (MathSciNet)
MR3302975

Zentralblatt MATH identifier
1344.58011

#### Citation

Mrowka, Tomasz; Ruberman, Daniel; Saveliev, Nikolai. Index theory of the de Rham complex on manifolds with periodic ends. Algebr. Geom. Topol. 14 (2014), no. 6, 3689--3700. doi:10.2140/agt.2014.14.3689. https://projecteuclid.org/euclid.agt/1513716053

#### References

• M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69
• M F Atiyah, I M Singer, The index of elliptic operators, I, Ann. of Math. 87 (1968) 484–530
• J B Conway, A course in operator theory, Graduate Studies in Mathematics 21, Amer. Math. Soc. (2000)
• J Dodziuk, de Rham–Hodge theory for $L^{2}$–cohomology of infinite coverings, Topology 16 (1977) 157–165
• M Farber, Topology of closed one-forms, Mathematical Surveys and Monographs 108, Amer. Math. Soc. (2004)
• R H Fox, A quick trip through knot theory, from: “Topology of $3$–manifolds and related topics”, (M K Fort, Jr, editor), Prentice-Hall, Englewood Cliffs, NJ (1962) 120–167
• B Hughes, A Ranicki, Ends of complexes, Cambridge Tracts in Mathematics 123, Cambridge Univ. Press (1996)
• S Kinoshita, On the Alexander polynomials of $2$–spheres in a $4$–sphere, Ann. of Math. 74 (1961) 518–531
• W Lück, $L^2$–invariants: Theory and applications to geometry and $K\!$–theory, Ergeb. Math. Grenzgeb. 44, Springer, Berlin (2002)
• J G Miller, The Euler characteristic and finiteness obstruction of manifolds with periodic ends, Asian J. Math. 10 (2006) 679–713
• J W Milnor, Infinite cyclic coverings, from: “Conference on the topology of manifolds”, (J G Hocking, editor), Prindle, Weber & Schmidt, Boston (1968) 115–133
• T Mrowka, D Ruberman, N Saveliev, An index theorem for end-periodic operators
• T Mrowka, D Ruberman, N Saveliev, Seiberg–Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant, J. Differential Geom. 88 (2011) 333–377
• A V Pajitnov, Proof of a conjecture of Novikov on homology with local coefficients over a field of finite characteristic, Dokl. Akad. Nauk SSSR 300 (1988) 1316–1320 In Russian; translated in Soviet Math. Dokl. 37 (1988) 824–828
• D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Berkeley, CA (1976)
• C H Taubes, Gauge theory on asymptotically periodic $4$–manifolds, J. Differential Geom. 25 (1987) 363–430