By Hajime Sato

The one so much tough factor one faces while one starts off to benefit a brand new department of arithmetic is to get a consider for the mathematical feel of the topic. the aim of this ebook is to assist the aspiring reader gather this crucial good judgment approximately algebraic topology in a brief time period. To this finish, Sato leads the reader via basic yet significant examples in concrete phrases. in addition, effects aren't mentioned of their maximum attainable generality, yet when it comes to the easiest and such a lot crucial circumstances. according to feedback from readers of the unique variation of this booklet, Sato has extra an appendix of necessary definitions and effects on units, basic topology, teams and such. He has additionally supplied references.Topics coated comprise basic notions resembling homeomorphisms, homotopy equivalence, basic teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute periods. gadgets and examples thought of within the textual content contain the torus, the Mobius strip, the Klein bottle, closed surfaces, phone complexes and vector bundles.

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**Example text**

Consequently, for every s ≥ s0 , we have s f (γ(s)) − f (γ(s0 )) = s0 s = d(f ◦ γ) du du (df )γ(u) (Xγ(u) )du s0 ≤ −ε0 (s − s0 ), so that lim f (γ(s)) = −∞, s→+∞ which is absurd. e Topology of the Sublevel Sets: When We Do Not Cross a Critical Value The topology of the level sets does not change as long as we do not cross a critical value. The same holds for that of the sublevel sets. Let V a = f −1 (]−∞, a]) denote the sublevel set of f for a. c, we said that if a is a regular value, then V a is a manifold with boundary.

The proof is based on the fact that all trajectories that are not stationary at a maximum end up at a minimum. Let c1 , . . , ck be the minima of f . The stable manifold W s (ci ) is diﬀeomorphic to an (open) interval; it consists of the two trajectories ending at ci and the point ci itself. In the closure Ai of this stable manifold there are (moreover) the starting points of these two trajectories. These starting points: • either are both maxima (in which case they can either coincide or not) • or at least one of them is a boundary point of V (in which case they are distinct).

We ﬁrst note that after replacing f by another function that is arbitrarily close in the C1 sense, if necessary, we may, and do, assume that f takes on distinct values at all of its critical points. Indeed, outside of Ω, we have (by compactness) df (X) < −ε0 for some ε0 > 0. We then choose a function h that is constant on each Morse chart Ωi , satisﬁes |dh| < 12 ε0 and for which f (ci ) + h(ci ) = f (cj ) + h(cj ) for i = j. The function f +h is still a Morse function, with the same critical points as f , and the vector ﬁeld X is still an adapted pseudo-gradient, but the critical values are now distinct.