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Beautiful & dynamic coastline and 5 blue spot background, solid color for window title, right click to control the sound volume, the ocean is the tide pool, with a gentle fountains or lapping water to give the images stability and create consistent environments.

Creating and designing themes is the pleasure of all those who are willing to join the design community of the creative world. Join TouchDesignz and start creating today, win the crowd and share your work by posting your comment and sharing the designs.Q:

Do the first two homotopy classes of complex curve vanishing cycles in the Milnor fiber of $f: \mathbb{C} \to \mathbb{C}$ are linearly independent?

Let $f:\mathbb{C} \to \mathbb{C}$ be a holomorphic map that has a single simple critical point at the origin. Then the Milnor fiber $M_f$ is the Milnor fiber of $f$ associated to the origin.
Let $e_1,\ldots,e_n$ be the irreducible components of $M_f$ and $[1],\ldots,[n]$ be the corresponding vanishing cycles. Then the first two homotopy classes of curves in $M_f$ are linearly independent. Can we prove that $n\geq 3$?
I am able to prove that $[1],[2],[3]$ are linearly independent but cannot extend to the rest of the homotopy classes.
This problem is discussed in this question here

A:

I don’t think that this is true. As a counterexample, consider the counterexample below (maybe it was already given there, but can’t see it right now):
$f(x) = x^2 + x + (x^2-x^2-1)^{1/2}$ which is holomorphic in the disk and for which $0$ is the only critical point (see Appendix A). The Milnor fiber is just a circle, and the vanishing cycle is of the same sign as the critical point. That is, the vanishing cycle is either positive or negative, and so the initial direction of the vanishing cycle is not in

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20 Sugared High Resolution 1920 x 1200 Images of SucculentIn addition to the 20 sugared images, we have also included 20 sugared windows images of 1920 x 1200 resolution.
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Minimum:
OS: Windows 10 64-bit
Processor: Intel i3-500 5300U or equivalent
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Recommended:
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Memory: 8 GB RAM
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