# Diffeomorphism type of the added sphere in simply connected surgery

A classical result of simply connected surgery theory, is that if two normal maps $$f:M_i\rightarrow X$$, $$i=0,1$$ are normally cobordant and if the dimension of the manifolds is odd, there exists a homotopy sphere $$\Sigma$$ such that $$M_0$$ is diffeomorphic to $$M_1\#\Sigma$$. This was first proved by Novikov in

Homotopically equivalent smooth manifolds. I, Izv. Akad. Nauk SSSR Ser. Mat., 28:2 (1964), 365–474. (Theorem 5.1)

but can also be found in Browder's

Surgery on simply-connected manifolds, Springer-Verlag (1972), Ergebnisse der Mathematik und ihre Grenzgebiete, Band 65 (II.3.7 Theorem)

and many other references of surgery theory.

My question is the following: is there a way to determine the diffeomorphism type of the added homotopy sphere?

• In dimension $4d-1$ it is sometimes possible to detect the diffeomorphism type of the added sphere using secondary invariants like (generalisations of) the Eells-Kuiper invariant. The question is then whether those invariants can be determined. – Sebastian Goette Jul 24 '19 at 18:40
• @SebastianGoette Do you know any reference where such secondary invariants are defined and with an example where this computation can be carried out? – Kafka91 Jul 25 '19 at 7:19
• You could start with the old papers by Kreck and Stolz, for example "Nonconnected moduli spaces of positive sectional curvature metrics," J. Amer. Math. Soc. 6 (1993), 825 – 850. Or, more recent, Crowley and Nordström, "The classification of 2-connected 7-manifolds, arXiv:1406.2226v2, 2018, where the inertia group can be nontrivial. – Sebastian Goette Jul 25 '19 at 22:27

The ambiguity in the the added homotopy sphere is captured by a suitable inertia group $$I(X)$$, which in this case is the group of homotopy spheres $$\Sigma$$ such that $$\Sigma$$ bounds a parallelizable manifold, and the standard homeomorphism $$\Sigma\# X\to X$$ is homotopic to the diffeomorphism.

There is a related inertial group $$\bar I(V)$$ that consists of homotopy spheres $$\Sigma$$ such that $$\Sigma\# X$$ and $$X$$ are diffeomorphic. For example, if $$X=S^3\times CP^2$$, then $$\bar I(X)=\Theta_7$$, the group of all homotopy $$7$$-spheres.

The group $$bP_d$$ of homotopy spheres bounding parallelizable $$d$$-dimensional manifolds is cyclic of known order. More precisely, the order is known except when $$d=126$$, the only remaining case of the Kervaire invariant problem, see here. Thus to understand the ambiguity in the the added homotopy sphere one needs to compute the index of $$I(X)$$ in $$bP_{d}$$ where $$\dim(X)=d-1$$.

Suppose $$d$$ is divisible by $$4$$ and $$d\ge 8$$, and $$X$$ is a closed oriented manifold of dimension $$d-1$$. Then $$bP_{d}$$ has large order given in terms of Bernoulli numbers. For such $$d$$ a theorem of L.Taylor says that the index of $$I(X)$$ in $$bP_{d}$$ is $$\ge 2$$. This is actually sharp, e.g., the index is $$2$$ for $$X=S^3\times CP^{2m}$$. The index is $$4$$ if $$X=S^7\times CP^2$$. On the other hand, Browder showed that $$I(X)$$ is trivial when $$d$$ is not divisible by $$8$$, the group $$H^1(X;\mathbb Z_2)$$ is zero, and $$X$$ stably parallelizable; thus in this case one can determine the diffeomorphism type of the added homotopy sphere".

More details and references to the above claims can be found here on p.8.

• Why do we need a bP sphere here? Is this because of the normal invariant? – Sebastian Goette Jul 24 '19 at 19:45
• @SebastianGoette: yes, summing with a non-bP sphere will change the normal cobordism class. – Igor Belegradek Jul 24 '19 at 20:22
• @IgorBelegradek: How can one see this, or on the other hand that a bP sphere does not affect the normal invariant? I am trying to show this by constructing a normal cobordism between the collapsing map M#Σ->M and the identity on M, but I don't see how to define the bundle over the obvious bordism Mx[0,1]#U where U is the parallelizable manifold bounding Σ. – Kafka91 Jul 25 '19 at 9:04
• @Kafka91: the proofs are in Browder's book. The original paper on surgery [Kervaire-Milnor "Groups of homotopy spheres"] is also an excellent source. Later textbooks all take this material for granted. – Igor Belegradek Jul 25 '19 at 11:23