Invariance of finiteness of K-area under surgery

K-area is an invariant for Riemannian manifolds introduced by Gromov as an obstruction to the existence of positive scalar curvature. However in general it is difficult to determine whether K-area is finite or not in spite of its natural definition. In this paper, we study how the invariant changes under surgery.


Introduction
The notion of K-area was introduced by Gromov [3]. It is an invariant for Riemannian manifolds with values in (0, +∞]. Roughly, K-area(M) measures how small C 0 curvature norms can be achieved for "non-trivial" vector bundles over a Riemannian manifold M. Here a "non-trivial" vector bundle E means a vector bundle with non-zero Chern numbers. Finiteness of K-area has a deep relationship with the existence of positive scalar curvature. The following theorem was proved by Gromov using the relative index theorem [2]. Theorem 1.1 [3] Let M be an even dimensional complete spin Riemannian manifold. If the scalar curvature Sc of M satisfies inf Sc > ε 2 , then K-area(M) ≤ cε −2 where c is a constant depending on the dimension of M.
In particular even dimensional spin manifold with K-area(M) = ∞ does not admit complete Riemannian metrics of uniformly positive scalar curvature.
Even though both notions of scalar curvature and K-area require Riemannian metrics, finiteness of K-area on a compact manifold depends only on its homotopy type. Hence infiniteness of K-area is a homotopical obstruction to the existence of positive scalar curvature on compact spin manifolds.
In this paper we verify the following. On the other hand the converse is easy to verify. Of course, the following Lemma 1.4 follows also from Theorem 1.2, but it can be verified without it. In fact, if M 1 has infinite K-area then there exists a "non-trivial" vector bundle E over M 1 with small C 0 curvature norm. Then we can construct another vector bundle over M 1 M 2 by extending E trivially onto M 2 .
We remark that the main theorem is analogous to the following.

Proposition 1.5 [1]
Let M be a compact manifold which carries a Riemannian metric of positive scalar curvature. Then any manifold obtained by surgeries in codimension ≥ 3 also carries a metric of positive scalar curvature.
The proof of our main theorem is rather different. The idea of the above proposition is that S q−1 × N admits a Riemannian metric of positive scalar curvature for q ≥ 3. On the other hand, we use a property that the cartesian product of spheres at the connecting region is simply connected. Any almost flat vector bundles over compact simply connected manifolds are trivial, which will be used to compute finiteness of K-area.
In [5] M. Listing studies so called "homology classes of finite K-area" and remarks that the homology of finite K-area in the dimensions lower than the largest one behave in the same way as the ordinary homology when taking the connected sums.
In [4] B. Hanke extends the concept of K-area by admitting Hilbert-A-module bundles of small or vanishing curvature. He defines the notion of infiniteness (and finiteness) of Karea of K -homology classes h ∈ K 0 (M) ⊗ Q for closed smooth manifolds M. It is shown that the K-area of the homological fundamental classes of area-enlargeable manifolds in the sense of [2] are infinite. Moreover he shows that oriented manifolds with fundamental classes of infinite K-area are essential. Manifolds are said to be essential if the classifying maps of universal covers map the homological fundamental classes to non-zero classes in the homology of the fundamental groups.

Definition and a fundamental lemma
Let E → M be a Hermitian vector bundle over a Riemannian manifold M, and let A be a section of * T M ⊗ End(E). Let us define where |A(ξ )| op denote the operator norm of A(ξ ) ∈ End(E).
Let K × (M) denote the isomorphism classes of Hermitian vector bundles equipped with compatible connections E = (E, ∇) over M, which satisfy the following conditions.  The following fundamental lemma is useful.

Lemma 2.2 Let M and M be Riemannian manifolds and let f : M → M be a smooth Lipschitz map of non-zero degree which is proper or constant outside a compact subset in
Lemma 2.2 implies that finiteness or infiniteness of K-area(M) is independent of the deformation of Riemannian metrics on compact subsets in M. In particular, the finiteness or infiniteness of K-area is a homotopy invariant of compact manifolds. This is stated in [3] without proof. We give a proof for convenience.
Here, we give some examples of K-area.

Example 2.3
(1) Let S 2m denote even dimensional spheres K-area(S 2m ) < ∞, which follows from Theorem 1.1. (2) If M be an oriented even dimensional closed simply connected manifold, then K-area(M) < ∞. Later in Lemma 3.2, every vector bundle (E, ∇) over a closed simply connected manifold with sufficiently small curvature R E,∇ < δ is topologically trivial, which implies that all Chern numbers of E are zero. Hence K-area(M) < 1 δ . K-area(S n ) < ∞ can be verified also from this.
Now consider an 2m-dimensional tori equipped with flat metrics g 0 which are induced by

Lemma 3.2 Let N be a compact simply connected Riemannian manifold and take E
Proof Fix a finite good open covering {V α } of N equipped with geodesic coordinates whose centers are p α . So each finite intersection of {V α } is contractible unless it is empty. Let E be a Hermitian vector bundle with R E < δ for some δ > 0. Fix an orthonormal basis be an orthonormal basis for E| p α obtained by the parallel transportation of e α 0 along γ α 0 , one of the minimal geodesics connecting p α 0 and p α . Extend e α on each V α by the parallel transportation along the geodesic t → exp p α (tv) where v is an unit tangent vector at p α .
Let ω α be the connection 1-form with respect to e α on V α . For x ∈ V α let γ x α be the (unique) geodesic connecting p α and x and for a piece-wise smooth curve γ let T γ be the parallel transportation along γ . Take x ∈ U α and X ∈ T x M. By the definition of e α , e α (exp where D t is a 2-dimensional disk whose boundary is the closed curve exp where c 1 is a constant depending on {V u a}. Keep in mind that constants c 1 , c 2 , · · · , c 6 which will appear below are independent of the vector bundle E. Let ψ βα : V α ∩ V β → U (r ) denote the transition functions, i.e., ψ βα e α = e β . By the definition of e α , ψ βα There exists a 2-dimensional disk D ⊂ N whose boundary is γ . By the compactness, we can take D so that area(D) < c 2 where c 2 is a constant depending on N and {V α }. Then we have By ψ βα e α = e β , we have dψ βα ⊗ e α + ψ βα ∇e α = ∇e β . Hence by (3.3) Taking into account the estimate (3.4) we can set ψ βα = exp(v βα ) for some v βα : V α ∩ V β → u(r ) using exp : u(r ) → U (r ) if δ > 0 is sufficiently small since exp is a diffeomorphism from a neighbourhood of 0 ∈ u(r ) to a neighbourhood of id ∈ U (r ). Remark that (3.4) and (3.5) implies v βα < c 2 δ and dv βα < 2c 1 δ (3.6) There exist open subsets W α and compact subsets K α such that W α ⊂ K α ⊂ V α and α W α = N . Note that these are independent of (E, ∇). Let {ρ α , ρ β , 1 − ρ α − ρ β } be a partition of unity associated to Construct an orthonormal frame e (2) on K α ∪ K β as follows; There is a constant c 3 > 0 such that |dρ α | < c 3 , |dρ β | < c 3 . Hence by (3.3), (3.6), and (3.7), This means the connection 1-form ω (2) associated to e (2) satisfies ω (2) (2) → U (r ) denote the transition function i.e., e γ = ψ γ (2) e (2) .
Repeat the above argument to construct a global orthonormal frameē for E which satisfies ∇ē < cδ. It means ω < cδ where ω is the connection 1-form with respect toē. Though c depends on N , it does not depend on (E, ∇).  Proof Choose ε 0 > 0 sufficiently small. For {0} × N and ε 0 , we can find δ = δ(ε 0 ) > 0 as in the preceding Lemma 3.2. Suppose that R E 0 < δ. Then we obtain a global orthonormal frame e for E 0 | {0}×N such that the connection 1-form ω 0 with respect to e satisfies ω 0 < ε 0 . Let E be a trivial Hermitian vector bundle without a connection on M (−2,6] which is an extension of E 0 . Extending e we obtain a orthonormal frame for E denoted also by e. Now compose a connection 1-form ω with respect to e on on (−2, 6] × N by where χ is a smooth function on (−2, 6] satisfying Since ω| (t,y) = ω 0 | y on (−2, 2) × N , the new connection denoted by ∇ can be patched with ∇ 0 .
. M is called a manifold obtained by p-surgery, or surgery in codimension q, along ϕ : S p × D q → M.
We assume that M is equipped with a Riemannian metric which coincides with the original one outside a compact neighborhood of (D p+1 × S q−1 ) ⊂ M .
Proof of Theorem 1.2 Since the finiteness of K-area is invariant under deformations of Riemannian metrics on compact subsets, we may assume that the "connecting region", the neighborhood of ∂(D p+1 × S q−1 ) ⊂ M is isometric to S p × (−4, 4) × S q−1 equipped with a canonical Riemannan metric.
Let E 0 = (E 0 , ∇ 0 ) be a Hermitian vector bundle equipped with a compatible connection. It is sufficient to verify that for sufficiently small δ > 0, R E 0 < δ implies that all Chern numbers of E 0 are zero.
Let f : M → M be a smooth Lipschitz map such that f = id outside S p × (−4, 4) × S q−1 , f (x, t, y) = (x, 0, y) for |t| < 2, and f * < 2. Consider f * E 0 , the pull-back of E 0 by f equipped with the induced connection f * ∇ 0 . Then R f * E 0 ≤ 2δ and the connection is invariant under the translation near the cylindrical boundary. Since deg( f ) = 1 the Chern numbers of f * E 0 are equal to those of E 0 .
Cut M along S p × {0} × S q−1 and remove D p+1 × S q−1 component. Let the resulting manifold be denoted by M .
In the case of p = 1, we can apply to f * E 0 | M the preceding Lemma 3.4 to obtain a vector bundle E = (E, ∇) over M (−2,6] with R E < ε which is trivial and flat on S p × (4, 6] × S q−1 . Remark that ∂ M = S p × {0} × S q−1 is simply connected by the condition q = 2.
Even in the case of p = 1, we claim that there exist a such extension of the vector bundle. In fact, consider the two copies of removed region D 2 × S n−2 and the vector bundle f * E 0 → (D 2 × S n−2 ) and reverse the orientation of one of them. We can patch them together along the boundary for the invariance of the connection of f * E 0 under the translation near the cylindrical boundary. Since the resulting manifold, the double of D 2 ×S n−2 , is homeomorphic to S 2 × S n−2 , by Lemma 3.2 there exists a global orthonormal frame e for the resulting vector bundle over S 2 × S n−2 such that the connection 1-form ω 0 with respect to e satisfies ω 0 < ε 0 . Hence there exists a such orthonormal frame for the restriction of f * E 0 onto a neighborhood of ∂ M . Then we can construct a vector bundle E = (E, ∇) over M (−2,6] with R E < ε which is trivial and flat on S p × (4, 6] × S q−1 in the same way as the Proof of Lemma 3.4. In the following argument the condition q = 2 is not needed. Deform the metric of S p × D q to have a product metric near the boundary S p × (−1, 1) × S q−1 so that it can be patched with M (−2,6] . The resulting manifold is homeomorphic to M. Since (E, ∇) is trivial and flat on S p × (4, 6] × S q−1 ⊂ (M (−2,6] ∪ S p × D q ), it can be extended on S p × D q trivially.
Let X be M \ M and let Y be M \ M . They are homeomorphic to D p+1 × S q−1 and S p × D q respectively. Glue X and (−Y ) together to compose a Riemannian manifold homeomorphic to S n where (−Y ) is the orientation reversed Y . Remark that ( f * E 0 , f * ∇ 0 ) on X and (E, ∇) on (−Y ) can be joined smoothly. Hence they define a Hermitian vector bundle equipped with a compatible connection (E, ∇) with a small curvature R < ε on X ∪ (−Y ).