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1.2 The Chas-Sullivan loop product
We denote by Λ(M) := C 0 (Ì, M) the space of free loops on M. Under the assumption that M
is an oriented n-dimensional manifold, it is possible to use the concatenation map Γ to define a
product of degree −n on H∗(Λ(M)). In order to describe the construction, we need to recall the
definition of the Umkehr map.
Let M be a (possibly infinite-dimensional) smooth Banach manifold, and let e : M0 ֒→ M be a
smooth closed embedding, which we assume to be n-codimensional and co-oriented. In other words,
M0 is a closed submanifold of M whose normal bundle NM0 := T M|M0/T M0 has codimension
n and is oriented. The tubular neighborhood theorem provides us with a homeomorphism 1 u :
U → NM0, uniquely determined up to isotopy, of an open neighborhood of M0 onto NM0,
mapping M0 identically onto the zero section of NM0, that we also denote by M0 (see [Lan99],
IV.§5-6). The Umkehr map is defined to be the composition
Hj(M) −→ Hj(M, M \ M0) ∼ =
−→ Hj(U, U \ M0) u∗
−→ Hj(NM0, NM0 \ M0)
τ
−→ Hj−n(M0),
where the first arrow is induced by the inclusion, the second one is the isomorphism given by
excision, and the last one is the Thom isomorphism associated to the n-dimensional oriented vector
bundle NM0, that is, the cap product with the Thom class τNM0 ∈ H n (NM0, NM0 \ M0). The
Umkehr map associated to the embedding e is denoted by
e! : Hj(M) −→ Hj−n(M0).
We recall that if M is an n-dimensional manifold, Λ(M) is an infinite dimensional smooth
manifold modeled on the Banach space C 0 (Ì,Ên ). The set Θ(M) of pairs of loops with the same
initial point (figure-8 loops),
Θ(M) := {(γ1, γ2) ∈ Λ(M) × Λ(M) | γ1(0) = γ2(0)},
is the inverse image of the diagonal ∆M of M × M by the smooth submersion
ev × ev : Λ(M) × Λ(M) → M × M, (γ1, γ2) ↦→ (γ1(0), γ2(0)).
Therefore, Θ(M) is a closed smooth submanifold of Λ(M)×Λ(M), and its normal bundle 2 NΘ(M)
is n-dimensional, being isomorphic to to the pull-back of the normal bundle N∆M of ∆M in M ×M
by the map ev × ev. If moreover M is oriented, so is N∆M and thus also NΘ(M). Notice also
that the concatenation map Γ is well-defined and smooth from Θ(M) into Λ(M). If we denote
by e the inclusion of Θ(M) into Λ(M) × Λ(M), the Chas-Sullivan loop product (see [ChS99]) is
defined by the composition
Hj(Λ(M)) ⊗ Hk(Λ(M)) ×
−→ Hj+k(Λ(M) × Λ(M)) e!
−→ Hj+k−n(Θ(M)) Γ∗
−→ Hj+k−n(Λ(M)),
and it is denoted by
o : Hj(Λ(M)) ⊗ Hk(Λ(M)) → Hj+k−n(Λ(M)).
We denote by c : M → Λ(M) the map which associates to every q ∈ M the constant loop q in
Λ(M). A simple homotopy argument shows that the image of the fundamental class [M] ∈ Hn(M)
under the homomorphism c∗ is a unit for the loop product: α o c∗[M] = c∗[M] o α = α for every
1If M admits smooth partitions of unity (for instance, if it is a Hilbert manifold) then u can be chosen to be a
smooth diffeomorphism.
2The Banach manifold Λ(M) does not admit smooth partitions of unity (actually, the Banach space C0 (Ì,Ên )
does not admit non-zero functions of class C1 with bounded support). So in general a closed submanifold of
Λ(M), or of Λ(M) × Λ(M), will not have a smooth tubular neighborhood. However, it would not be difficult to
show that the submanifold Θ(M) and all the submanifolds we consider in this paper do have a smooth tubular
neighborhood, which can be constructed explicitly by using the exponential map and the tubular neighborhood
theorem on finite-dimensional manifolds.
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