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?

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    $\begingroup$ 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. $\endgroup$ – Sebastian Goette Jul 24 '19 at 18:40
  • $\begingroup$ @SebastianGoette Do you know any reference where such secondary invariants are defined and with an example where this computation can be carried out? $\endgroup$ – Kafka91 Jul 25 '19 at 7:19
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    $\begingroup$ 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. $\endgroup$ – 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.

  • $\begingroup$ Why do we need a bP sphere here? Is this because of the normal invariant? $\endgroup$ – Sebastian Goette Jul 24 '19 at 19:45
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    $\begingroup$ @SebastianGoette: yes, summing with a non-bP sphere will change the normal cobordism class. $\endgroup$ – Igor Belegradek Jul 24 '19 at 20:22
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    $\begingroup$ @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. $\endgroup$ – Kafka91 Jul 25 '19 at 9:04
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    $\begingroup$ @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. $\endgroup$ – Igor Belegradek Jul 25 '19 at 11:23

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