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Let $X\to D$ be a proper $C^\infty$ map with $D$ an open disk about the origin in some Euclidean space. Suppose $0\in D$ is the only singular value, i.e that over $D^\times=D\setminus \left\{ 0 \right\}$ the map is a submersion. A choice of Riemannian metric on $X$ furnishes a notion of gradient. Flowing along $-\nabla\|f\|^2_2$ furnishes a map $X\to f^{-1}(0)$ pushing the generic fibers radially inwards into the special fiber (this tacitly requires properness). Precomposition with inclusion of a generic fiber defines the specialization map for any $t\in B^\times$ from a generic fiber to the special fiber.$$\psi_t:f^{-1}(t)\to f^{-1}(0)$$

Question 1. It is geometrically intuitive that specialization maps with respect to different Riemannian metrics are homotopic. Is this true? How to prove it?

Remark. It's been asked here whether the specialization map is surjective. Nobody's answered yet!

According to MO and many other sources,

A vanishing cycle is a homology class of the generic fibre that shrinks to zero in the special fibre.

"Shrinking in the special fiber" seems to suggest following the specialization map. It seems an elementary interpretation of a vanishing $n$-cycle is an element in the kernel of $\pi_n(\psi_t)$ or $\mathrm H_n(\psi_t)$, i.e an $n$-cycle that dies upon specialization. Similarly, an emerging $n$-cycle would be an element of the associated cokernel (or at least complement of the image).

Question 2. In this paper by Meigniez, the author gives a direct definition of (representative of) vanishing and emerging cycles, but seems to have the reverse convention for direction. He writes on page 2 "when moving from a given fiber to immediately neighboring ones over some path in the base", which is opposite from the direction of specialization. Is this convention used elsewhere?

Question 3. Where are emerging cycles in the literature? Alternatively, what's the "role played by vanishing cycles" as opposed to emerging cycles?

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  • $\begingroup$ Is the map always defined? Cannot the gradient flow sort of smear out when approaching the special fiber? In principle its trajectories may fail to have single limit in the special fiber... $\endgroup$ Mar 25 '20 at 19:55
  • $\begingroup$ @მამუკაჯიბლაძე I think the properness hypothesis precludes smearing out, but maybe I am wrong. $\endgroup$
    – Arrow
    Mar 26 '20 at 12:50

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