Spectral sequences

ΣSⁿS^{n 1} . This and result (5.9) are then used to prove that

HpΩS^{n 1}q R u

with |u| n , (5.10)

where u is represented by an explicit cycle of loops inS^{n 1}, cf. section 4.3.

A more systematic method to compute (co-)homology groups and certain products also for free loop spaces is provided by spectral sequences which are briey discussed in the next section.

k₁peq A₂ :i₁pA₁q ^{i2:i1|i1pA1q //}A₂

j2

uu

i₁paq

6

zzres

ff

E₂ :HpE₁, d₁ :j₁k₁q

k2

jj

rj₁paqs . Deriving exact triangles from given ones can be done innitely often.

A spectral sequence is a sequence of dierential complexes pE_r, d_rq withE_{r 1} HpE_r, d_rq.

It stabilizes if E_{l 1} E_{l 2} :E₈ and converges to HpKq if E₈ à

p

HpKqp{HpKqp 1

for the induced cohomology ltrationHpKq HpKq0 HpKq1 HpKq2  given byHpKqp : piq^pHpKpq. If HpKq is a vector space over a eld kwe have

à

p

HpKqp{HpKqp 1 HpKq . Theorem 5.6 (e.g. Theorem 14.6. in [2])

If the ltration has nite length l_n, that is

Kⁿ K₀ⁿK₁ⁿ K_lⁿ_n K_lⁿ_{n 1} 0

for each dimension n ¥0, the induced spectral sequence stabilizes and converges.

We are in the situation required in the theorem when considering a double (bigraded) complex

K à

p,q¥0

K^p,q with dierentials

δ:K^p,q ÑK^{p 1,q} and d¹ :K^p,q ÑK^{p,q 1} such that pd¹q² 0,δ² 0and d¹ δ δd¹.

It yields a single graded ltered complex pK À

n¥0

Kⁿ, Dq with Kⁿ : À

p qn

K^p,q and ltration K_p0 : À

q¥0, i¥p0

K^i,q of nite length in each dimension. The denitions are best illustrated as in gure 5.1.

Finite length is given since K_pⁿKⁿXK_p 0forp¡n. For the E₁-page pE₁, d₁q we get

E1 à

p¥0

HpKp{Kp 1, Dq à

p,q¥0

HpK^p,q, d¹q : à

p,q¥0

E₁^p,q

115

since δ|Kp{Kp 1 0. For d₁ j₁k₁ :HpK_p{K_{p 1}q ÑHpK_{p 1}{K_{p 2}q we get ras ÞÑ rDas rδas

since k₁ is the connecting homomorphism of the long exact sequence and thus E₂^p,q H^ppH^,qpK, dq, δq.

This principle is manifested in the zig-zag Lemma as described in [2].

Lemma 5.7 (Ÿ14 of [2]) For x₀ PK^p,q one has

rx₀sk 1 PE_k^p,q₁ ô Dk-zig-zag px₀, ..., x_kq

i.e. dx₀ 0, δx_l p1q^{p 1}dx_{l 1} pl kq and further

d_{k 1}rx₀sk 1 rδx_ksk 1 PE_k^{p k}₁ ^1,qk .

Our motivation for studying spectral sequences are (co-)homological computations for brationsF ãÑE Ñ^π B forF, E, B being CW-complexes andB being path-connected.

ForU tU_αuαPI being a cover of B we dene a double complex K^{p 1,q} ÐÝ^δ K^p,qÝÑ^d K^{p,q 1} ,

with K^p,q : C^ppπ¹pUq, C^qq being the p-th ƒech cochain group with values in the presheaf of singular q-cochains. This set-up yields a spectral sequence. As described in the literature, if π₁pBq acts trivially onH^qpFq we have

H^qpπ¹pUqq H^qpFq

if U is a good cover of B, that is it is locally nite and non-empty intersections U_α1 X XU_αr are dieomorphic toRⁿ.

Following chapter 5 of [30] for the corresponding spectral sequence we get:

116

• E₂^p,q H^ppU , H^qpFqq

• pE_r, d_rq converges to HpEq

• The universal coecient theorem yields

E₂^p,q H^ppBq bH^qpFq

if we use eld coecients and H^qpFq is nite dimensional for all q.

Analogously we could work with chains instead of cochains and would get the same statements for homology. For the cup product on cohomology or the loop product on HpLXq, the statement generalizes in a way such that module isomorphisms become algebra isomorphisms. For this we refer to [10] and [30]

As an example considerS⁸ ÑCP⁸ as a realization of the universal S¹-bundle. Con-tractibility ofS⁸ yields

E_k^p,q_¥3 E₈^p,q

"

0 , pp, qq p0,0q k , pp, qq p0,0q

when using coecients in a eldk. This follows by degree reasons whereas theE2-page is given by

which implies

HpBS¹;kq HpCP⁸;kq krxs with |x| 2.

The class x P H²pCP⁸;kq is known as the Euler class which can be easily dened for general sphere bundles using spectral sequences. The exactness of the previously mentioned Gysin sequence is also straightforward.

Both statements can be seen as follows:

In general for a brationF ãÑE ÑB an element ωP HⁿpFq is called transgressive if d₂pωq d_npωq 0.

Since p, q ¥ 0 we have d_{k¥n 2}pωq 0 by degree reasons. In this situation, the map ωÞÑd_{n 1}pωq is called the transgression map.

117

E₂^0, HpSⁿ;kq krωs{pω²q .

Thus by degree reasons E2 En 1 and En 2 E₈. So for computing HpEq we just need to understand

d_{n 1}pωq :ePH^{n 1}pBq,

called the Euler class of the bundleE ÑB. We immediately get that a trivial sphere bundle has a vanishing Euler class.

In total the dierential d_{n 1} onE_{n 1} is given by

H^ppBq bHⁿpSⁿq ÝÑH^{p n 1}pBq bH⁰pSⁿq xbω ÞÝÑ pxYeq b1 .

For coecients in a eld it yields HpEq kerp Yeq `HpBq{imp Yeq which may be interpreted as

ÑHⁱpEqÑ^π HⁱⁿpBqÑ^Ye H^{i 1}pBqÑ^π H^{i 1}pEq Ñ

where π is the projection to kerp Yeq and π : HpBq Ñ HpBq{imp Yeq. This is the already mentioned Gysin sequence that is clearly exact.

118

Bibliography

[1] Basu, S.: Transversal String Topology & Invariants of Manifolds. Stony Brook University, PhD thesis, August 2011.

[2] Bott, R. and L. W. Tu: Dierential Forms in Algebraic Topology. Springer-Verlag New York, 1982. volume 82.

[3] Carlsson, G. and R. J. Milgram: Stable Homotopy and Iterated Loop Spaces.

North Holland, 1995. in I. M. James, editor, Handbook of Algebraic Topology.

[4] Carmo, M. P. do: Riemannian Geometry. Birkhäuser Boston, 1992.

[5] Chas, M. and D. Sullivan: String Topology. arXiv:math/9911159 [math.GT], 1999.

[6] Chataur, D. and A. Oancea: Basics on free loop spaces. in 'Free Loop Spaces in Geometry and Topology' (J. Latschev, A. Oancea, eds.), to appear in European Mathematical Society Publishing House as IRMA Lectures in Mathematics and Theoretical Physics Vol. 24.

[7] Cieliebak, K.: Lectures on String Topology. Universität Augsburg, lecture notes, 2013.

[8] Cieliebak, K., K. Fukaya and J. Latschev: Homological algebra related to surfaces with boundary. arXiv:1508.02741 [math.QA], 2015.

[9] Cohen, R. L., K. Hess and A. Voronov: String Topology and Cyclic Homol-ogy. Birkhäuser Verlag, 2006. Advanced Courses in Mathematics CRM Barcelona.

[10] Cohen, R. L., J. D. Jones and J. Yan: The loop homology algebra of spheres and projective spaces. Progress in Mathematics, Volume 215, Number 2:7792, 2003.

[11] Cohen, R. L. and J. D. S. Jones: A homotopy theoretic realization of string topology. Math. Ann. 324, no. 4:773798, 2002.

[12] Eliashberg, Y., A. Givental and H. Hofer: Introduction to Symplectic Field Theory. Geom. Funct. Anal., Special Volume:560673, 2000.

[13] Fukaya, K.: Application of Floer homology of Langrangian submanifolds to sym-plectic topology. Springer, in: Paul Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology:231276, 2006.

119

Press, 2009. vol. 46.

[15] Gleason, A. M.: Spaces with a compact Lie group of transformations. Proceed-ings of the American Mathematical Society, 1:3543, 1950.

[16] Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math.

82, no. 2:307347, 1985.

[17] Hansen, V. L.: On the Fundamental Group of a Mapping Space. Compositio Mathematica, 28 Fasc. 1:3336, 1974.

[18] Hatcher, A.: Algebraic Topology. Cambridge University Press, 2002.

[19] Hatcher, A.: Vector Bundles & K-Theory. http://www.math.cornell.edu/

~hatcher/VBKT/VBpage.html, 2009.

[20] Irie, K.: Transversality problems in string topology and de Rham chains.

arXiv:1404.0153v2 [math.GT], 2014.

[21] Kadeishvili, T. V.: On the homology theory of bre spaces. Russian Math.

Surveys, 35:231238, 1980.

[22] Keller, B.: Introduction to A-innity Algebras and Modules. Homology, Homo-topy and Applications, 3, No. 1:135, 2001.

[23] Kosinski, A. A.: Dierential Manifolds, vol. 138. Academic Press, 1992. Pure and Applied Mathematics.

[24] Kriz, I. and J. P. May: Operads, Algebras, Modules, and Motives. Astérisque, no. 233, 1995.

[25] Latschev, J.: Fukaya's work on Lagrangian embeddings. in 'Free Loop Spaces in Geometry and Topology' (J. Latschev, A. Oancea, eds.), to appear in European Mathematical Society Publishing House as IRMA Lectures in Mathematics and Theoretical Physics Vol. 24.

[26] Lee, J. M.: Introduction to Topological Manifolds, vol. 218. Springer, 2010.

Graduate Texts in Mathematics.

[27] Loday, J.-L. and B. Vallette: Algebraic Operads, vol. 346. Springer, 2012.

Grundlehren der mathematischen Wissenschaften.

[28] Massey, W. S.: Algebraic Topology: An Introduction, vol. 56. Springer, 1977.

Graduate Texts in Mathematics.

[29] May, J. P.: A Concise Course in Algebraic Topology. University Of Chicago Press, 1999.

[30] McCleary, J.: A User's Guide to Spectral Sequences. Publish or Perish, Inc., 1985. Mathematics Lecture Series 12.

120

[32] Milnor, J.: On Spaces Having the Homoptopy Type of a CW-Complex. Trans-actions of the American Mathematical Society, 90:272280, 1959.

[33] Ratcliffe, J. G.: Foundations of Hyperbolic Manifolds (second edition), vol.

149. Springer, 1994. Graduate Texts in Mathematics.

[34] Silva, A. C. da: Lectures on Symplectic Geometry. Springer, 2008. 2nd edition.

[35] Thom, R.: Quelques propriétés globales des variétés diérentiables. Comment.

Math. Helv., 28:1786, 1954.

[36] Wilson, S. O.: On the Algebra and Geometry of a Manifold's Chains and Cochains. Dissertation, Stony Brook University, 2005.

121

Hiermit erkläre ich an Eides statt, dass ich die vorliegende Dissertationsschrift selbst verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.

Hamburg, den 6. November 2016

Johannes Huster

Im Dokument Contributions to the string topology of product manifolds (Seite 118-128)