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. though the definition of K-area is quite natural. In this paper, we study how the invariant changes under surgery.

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 0.5.  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 [Li10] 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 [Ha11] 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 K-area 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  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. (i) (E, ∇) are isomorphic to the trivial bundles C r equipped with flat connections outside compact subsets of M . (ii) (E, ∇) have a non-zero Chern number. i.e. there exists a (multivariable) polynomial p such that Gr96]). Let M be an even dimensional Riemannian manifold and let R = R E = R E,∇ denote the curvature tensor of (E, ∇). Then K-area of M is defined by K-area(M ) = ∞ if and only if for any ε > 0, there exists a vector bundle (E, ∇) ∈ K × (M ) with a small curvature R < ε.
The following fundamental lemma is useful. Lemma1.2 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 [Gr96] without proof. We give a proof for convenience.
Consider the vector bundle f * E → M equipped with the induced connection f * ∇. Since f is proper or constant outside a compact subset, f * E is isomorphic to a flat bundle C r outside a compact subset. Moreover, Here, we give some examples of K-area.
(2) If M be an oriented even dimensional closed simply connected manifold, then K-area(M ) < ∞. Later in lemma 2.2, 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. (3) K-area(T 2m ) = ∞ where T 2m denote even dimensional tori. It follows from theorem 0.1 that T 2m and hence T 2m−1 do not admit Riemannian metrics of positive scalar curvature. proof of (3). Generally let M = (M, g) be a Riemannian manifold equipped with a metric g. Observe that K-area(M, c 2 g) = c 2 K-area(M, g) by the preceding lemma 1.2.

Surgery
Let M 1 and M 2 be Riemannian manifolds and let M 1 ♯M 2 denote the connected sum of M 1 and M 2 equipped with a Riemannian metric which coincides with the original metric of M 1 ⊔ M 2 outside a compact neighborhood of the connecting region.
Example 2.1. Let M be a 2m dimensional closed spin manifold. Then T 2m ♯M does not admit a Riemannian metric of positive scalar curvature. In fact K-area(T 2m ) = ∞ implies K-area(T 2m ♯M ) = ∞ and apply theorem 0.1.
However, the converse of lemma 0.4 is not trivial. The following two lemmata are used to verify theorem 0.2.

Lemma 2.2. Let N be a compact simply connected Riemannian manifold and take
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| pα 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 pα (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 x (tX)) = T γ exp x (tX) 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.
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, ∇).
Remark 2.3. The proof of lemma 2.2 also holds if N is not connected but each connected component is simply connected by applying the arguments on each connected component.
Lemma 2.4. Let M be a Riemannian manifold with a simply connected boundary N = ∂M , and let E 0 = (E 0 , ∇ 0 ) be a Hermitian vector bundle over M equipped with a compatible connection. Suppose that a neighborhood of ∂M is equipped with a product metric of (−2, 2] × N and that the connection ∇ 0 restricted to (−2, 2] × N is invariant under the translation.
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 2.2. 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 .
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 0.2. 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 2.4 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 2.2 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 2.4 . 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 ).