Kusner's manifold exercises (Math 704, Spring 2000) All material is Copyleft 2000 by Rob Kusner* ======================================================================== 1/27 Compute the fundamental group of a circle, of a figure 8, and of a torus (circle times a circle). ------------------------------------------------------------------------ Let V be an n-dimensional vector space over R. Compute the dimension of the exterior algebra V. [Hint: if {e1, e2, ... , en} is a basis for V, show that {1, e1, e2, ... , en, e1e2, ... , en-1en, ... , e1e2e3...en} is a basis for the exterior algebra. ======================================================================== 2/1 From the above problem, you should have found dimension of the degree k vectors (or k-vectors, for short) in the exterior algebra of V to be n \choose k = n!/k!(n-k)! Since n \choose k = n \choose n-k, there is an isomorphism between k- and (n-k)-vectors. Here's a way to define such an isomorphism, "Hodge duality": if {e1, e2, ... , en} is a basis for V, then the n-vectors e1e2...en and -e1e2...en are ORIENTATIONS for V; pick an orientation (say, the + one); then any k-vector is paired with its Hodge dual (n-k)-vector according to this rule for basis vectors: *(ei1ei2...eik) := (+/-) e1...ei1^...eik^...en where the v^ means v is omitted, and where the (+/-) is chosen to make *(x)x a nonnegative multiple of the (+) orientation chosen, i.e. *(ei1ei2...eik)ei1ei2...eik = e1e2...en. Show that, on k-vectors, *^2 = (+/-)Id - how does the sign depend on k? In case V = R^2 and k=1, *^2 = -Id, and * is "rotation by 90 degrees". In case V = R^4 and k=2, *^2 = Id; compute the (+/-)1 eigenspaces for *, i.e. the self- and anti-self-dual 2-vectors on R^4. ------------------------------------------------------------------------ If L:V -> V is a linear map, then L induces a linear map on the exterior algebra. In particular, show that the induced map on the 1-dimenional space of n-vectors is multiplication by det(L). ------------------------------------------------------------------------ If V* is the dual space of V, i.e. the space of all linear maps from V to R, then we can define exterior algebra of V* as well. Show that the k-vectors on V* can be regarded as skew-symmetric k-linear maps from V to R (or k-forms, for short). Show that L as above induces a linear map on k-forms - what is this map? ======================================================================== 2/3 We have seen how to define Hodge duality for smooth forms: extend it to commute with multiplication by smooth functions. For example, let \theta = (x dy - y dx)/(x^2 + y^2) be a 1-form on R^2 \ {0}; then *dx = -dy and *dy = dx extend to *\theta = (x dx + y dy)/(x^2 + y^2) = dr/r = d(ln r) where r^2 = x^2 + y^2. Compute d\theta and d(*theta). Interpret \theta in polar coordinates. Do these forms remind you of anything from complex analysis?! ----------------------------------------------------------------------- The differential operators grad, curl and div from advanced calculus have a nice interpretation in terms of exterior derivative d and Hodge duality *, as follows. Let I, J and K be the standard constant vector fields on R^3. Then any vectorfield A = aI + bJ + cK on R^3 is determined by smooth functions a, b and c. We may identify A with a 1-form #A = a dx + b dy + c dz (here # or "sharp" is what Marcel Berger calls a "musical isomorphism" - its inverse should be "flat", but I will use # for it also in what comes next, so ## = Id). Considering the spaces of forms on R^3, the operator d, and Hodge *: <---*---> \Omega^0 -d-> \Omega^1 -d-> \Omega^2 -d-> \Omega^3 <---------------*-----------------> please check the following (up to sign): grad f = #df curl A = #(*d(#A)) div A = *(d*(#A)) and in particular, since d^2 = 0, we have the calculus facts curl(grad f) = 0 and div(curl A) = 0. ======================================================================= 2/8 We observed that the kernel of a 1-form \alpha on R^n can be viewed as a field of (n-1) dimensional planes on R^n. If \alpha is EXACT, that is, if \alpha = df for some function (0-form) f on R^n, these planes will be tangent planes to the level sets of f. Every 1-form on R^1 is exact. (Hint: Fundamental Theorem of Calculus.) In general, there is no such f (even locally). Poincare's Lemma asserts that if \alpha is CLOSED (d\alpha = 0) on R^n, then it is exact. (Hint: FTOC, in the guise of line integrals and Stokes' Theorem.) ----------------------------------------------------------------------- On R^2 there are plenty of non-exact 1-forms \alpha (x dy, for example), but we can always find a pair of nonvanishing functions f and g so that g \alpha = df and thus ker(\alpha) is still tangent to the curves {f = constant}. (In ODE classes, g is called a "multiplying factor".) Consider the 1-form \alpha = x dy - dz on R^3. Sketch the field of planes on R^3 defined by ker(\alpha). Compute d\alpha to conclude there are no level surfaces {f = constant} for which these are tangent planes. Is there ANY family of surfaces to which these planes are tangent? (That is, can you find the analog of a multiplying factor g as above?) ======================================================================= 2/11 The last of the above problems should remind you of the Frobenius integrability theorem. If \alpha1, ... , \alphak are (pointwise) linearly independent 1-forms on R^n (i.e. their k-fold wedge product is non-vanishing), then the intersections of their kernels defines a codimension-k distribution on R^n. The distribution is integrable provided the derivatives of the \alpha's are 2-forms lying in the ideal generated by the \alpha's: for each i = 1, ... , k d\alphai = \alpha1\thetai1 + ... + ak\alphak\thetaik (F) for some 1-forms \thetai1, ... , \thetaik. --------------------------------------------------------------------- To see this condition agrees with the version using vector fields, first show that the exterior derivative of a 1-form \omega applied to a pair of general vector fields X and Y satisfies d\omega(X,Y) = X\omega(Y) - Y\omega(X) - \omega([X,Y]). (Hint: the "correction term" involving Lie bracket [X,Y] is needed to make both sides bilinear with respect to smooth functions.) ---------------------------------------------------------------------- On R^3, the distribution defined by ker(\alpha) is integrable iff \alpha d\alpha = 0. ======================================================================= 2/15 Conversely, a distribution defines an ideal I generated by the linear space of 1-forms I(1) which annihilates the distribution. (If the \alpha's are given in advance, their span - with smooth function coefficients - defines I(1), and I is as above.) ----------------------------------------------------------------------- To check that condition (F) above agrees with vector field version of Frobenius condition - namely, if X and Y are tangent to the distribution, then so must their Lie bracket [X,Y] - we need to know that a 2-form \Omega can be written as \omega \theta if and only if \Omega(X,Y) = 0 for all X, Y in ker(\omega). (The "only if" direction is clear - please check the "if" direction.) ======================================================================= 2/18 Although we don't have class today, let me give you a computation to work through. If we insist upon the Leibniz rule d(\al \om) = d\al \om + (-1)^deg(\al) \al d\om the above formula for the exterior derivative of a 1-form \om applied to a pair of vector fields X and Y, along with the formula for exterior derivative of a function f df(X) = Xf d\om(X,Y) = X\om(Y) - Y\om(X) - \om([X,Y]) has a unique extension to k-forms. For example, consider a 2-form \Om = \al \om (the general 2-form is a sum of these SIMPLE 2-forms, though in R^3 observe that all 2-forms are simple); then we must have d\Om(X,Y,Z) = (d\om \al)(X,Y,Z) - (\om d\al)(X,Y,Z) = d\om(X,Y) \al(Z) - \om(X) d\al(Y,Z) + [other permutations of X, Y and Z] = [a formula which uses only the definition of exterior derivative d on lower degree forms] which suggests what d of a k-form applied to X0, X1, ... , Xk is. ======================================================================= 2/22 Determine all smooth maps F:R -> R such that F*(dx) = dx. Describe all smooth maps F:R^2 -> R^2 such that F*(dx dy) = dx dy. Suppose F: R^2 -> R^2 is defined by F(x,y) = (x cosy, x siny). Compute the pullbacks F*(dx), F*(dy) and F*(dx dy). Suppose F: R^3 -> R^3: (x,y,z) -> (x cosy sinz, x siny sinz, x cosz). Compute the pullback F*(dx dy dz). Let F: R^n -> R^m be a smooth map. Verify that pullback commutes with exterior derivatives and products, that is F*\alpha F*\omega = F*(\alpha \omega) d(F*\alpha) = F*(d\alpha) for any k-form \alpha and p-form \omega on R^m. ======================================================================= 2/25 Identify R^2 with C via x1 + i y1 = z1, and similarly identify R^4 with C^2. Let \omega = dx1 dy1 + dx2 dy2 be a symplectic form on R^4. Express \omega in terms of dz's and dz_bar's. Which rotations of R^4 preserve \omega, that is, which A in SO(4) satisfy A*(\omega) = \omega? Think of the first coordinate subspace C in C^2 as defining an embedding F: R2 -> R^4, with F*(\omega) = dx1 dy1. Discuss how (AF)*(\omega) = c dx1 dy1 depends on the rotation A. For instance, if A is unitary, c = 1; for what A will c = 0 (that is, which A rotate C = R^2 into a Lagrangian plane in C^2 = R^4)? ======================================================================= 2/29 Upto diffeormorphism, there are two real line bundles over S^1: the trivial bundle S^1 x R, and the M\"obius strip. How many R-bundles are there over S^2? How many over S^1 x S^1? Investigate the R^2 bundles over S^2 (see Hirsch's _Differential Topology_ for how to classify R^k-bundles over S^n). ======================================================================= 3/3 The tangent bundle TS^2 is nontrivial (consider the complex charts with coordinate change z=1/w, and compute effect on vectors along the equator |z|=1=|w|). The tautological complex line bundle Taut(S^2) over S^2 is the R^2-bundle whose unit vectors give the Hopf circle bundle S^3 -> S^2. Viewing S^2 = CP^1, the fiber over [z0:z1] is the C-line in C^2 spanned by the vector (z0,z1). Compare these with TS^2 and UTS^2. ======================================================================= 3/7 [From Spivak's _Comprehensive Introduction to Differential Geometry_, volume 1, appendix A] Let M be a topological n-manifold, that is, a Hausdorff space such that each point of M has a neighborhood homeomorphic to R^n; then the following are equivalent: i. (Each component of) M has a countable base of open sets; ii. M is metrizable; iii. M is paracompact, that is, any open cover of M has a locally finite refinement. ----------------------------------------------------------------------- Try to fill in the details for constructing the partition of unity subordinate to a (locally finite) cover of a manifold by charts, using the characteristic function of the (slightly shrunken) image of a chart (and convolution) to contruct the smooth bump function supported inside the chart. ======================================================================= 3/10 We computed the deRham cohomology H*(S^1) using the Mayer-Vietoris exact sequence for the cover of S^1 by two (interval) charts. Use this as the beginning of an induction to compute H*(S^n), where S^n is again covered by N- and S-polar charts. [Hint: argue - informally, for now - that the intersection of these charts has the same deRham cohomology as S^{n-1}] ======================================================================= 3/21 We showed that the "connecting homomorphism" H^k(C) -d*-> H^k+1(A) in the long exact cohomology sequence (associated to a short exact sequence 0 -> A -f*-> B -g*-> C -> 0 of co-complexes) is well-defined, as are H^k(A) -f*-> H^k(B) and H^k(B) -g*-> H^k(C), and we checked exact exactness at H^k(B). Also check exactness at H^k(A) and H^k(C). ----------------------------------------------------------------------- Explicitly describe the above homomorphisms in the case of the Mayer-Vietoris exact sequence. ======================================================================= 3/23 (Thursday meeting) Here's another way to think about orientatability and orientations: Every manifold M has 2-fold covering space M~ which is orientable (in the sense that it has an atlas of consistent charts). [Hint: for each chart on M, there is a pair of charts on M~, related by the reflection map we called R in class.] Assume M is connected. M is orientable iff M~ is disconnected. (In fact, M~ will have exactly two components, each corresponding to a choice of orientation on M.) Consequently, any simply connected manifold is orientable. Suppose now that M is the M\"obius band. What is the orientation double covering M~? Deduce from this that the M\"obius band is not orientable. ======================================================================= 3/27 We discussed the interior product (or contraction) of a vector field X with a k-form \omega: \i_X\omega is the (k-1)-form satisfying \i_X\omega(X-1,...,X_k-1) = \omega(X,X_1,...,X_k-1) for any vector fields X_1,...,X_k-1. Explore the properties of interior product. In particular, show that \i_Xd\omega + d\i_X\omega = d/dt|_t=0(\Phi_t*\omega) =: L_X\omega the Lie derivative of \omega with respect to the vector field X (here \Phi_t is the flow with d/dt|_t=0(\Phi_t) = X). ---------------------------------------------------------------------------- Let \omega be a volume form (i.e. non-vanishing n-form) and let X be a vector field on M. The divergence of X is the function div X defined by L_X \omega = div X \omega. Check that this agrees with the usual definition of divergence on R^n. For general M, show that the flow \Phi_t of X is volume preserving (\Phi_t*\omega = \omega) iff div X = 0. ----------------------------------------------------------------------- Suppose M also has boundary \delM. We know how to induce an orientation on \delM using an outward pointing vector field along \delM; but in order to get a volume (n-1)-form \alpha on \delM from the volume n-form \omega we need to choose a particular vector field Y along \delM and form the interior product \alpha = \i_Y\omega|_\delM. If M has a Riemannian metric <.,.>, then a canonical choice of Y is the outward unit normal vector field along \delM. Use Stokes' Theorem to derive the Divergence Theorem: \int_M div X \omega = \int_\delM \alpha ----------------------------------------------------------------------- Formulate and prove a version of Stokes' Theorem for a manifold "with corners", whose charts correspond to open subsets of a polyhedron in R^n. (Note that a manifold with boundary is a special case, where the polyhedron is the upper-half-space H^n in R^n.) ======================================================================= 3/31 What are the homotopy classes of maps from the zero sphere S^0 = {-1,+1} to a manifold M? In particular, how is this related to the zeroth deRham cohomology group H^0(M)? ----------------------------------------------------------------------- Show that x -> x/|x| defines a deformation retraction from R^n+1\{0} to S^n. Is there a deformation retraction from B^n+1 to S^n? ======================================================================= 4/4 Verify that the existence of a chain homotopy K between two chain maps f and g (i.e. f - g = (+/-) Kd + (+/-)dK) means that f* = g* on (co)homology. Is the converse true? ----------------------------------------------------------------------- Recall the trivial line bundle \pi: M x R -> M and its zero section \sigma: M -> M x R. Use the fact that \pi* and \sigma* are inverses on cohomology to deduce the Poincare' Lemma. [Take M = R^n-1 and use induction on n from the known case n=1.] ======================================================================= 4/6 (Thursday) Check the signs in the chain homotopy K (integration along the fibre R) which we constructed between \pi*\sigma* and the identity on \Omega*(M x R). ----------------------------------------------------------------------- Try to understand Bott-Tu's argument for the compactly supported case. Note that the chain homotopy K is a little bit different here. ======================================================================= 4/11 We defined the (mod 2 and the) integer degree for a proper smooth map f: M -> N between orientable (or orientable mod 2 - this is an empty condition, since any manifold is mod 2 orientable) manifolds of the same dimension to be the (unsigned and) signed sum over the preimage of any regular value p in N. Check that for connected N, these are well defined (i.e. independent of p). ----------------------------------------------------------------------- For manifolds of different dimensions, there are generalizations of the degree. For instance, the Hopf invariant H(f) is an integer defined for any map f: S^3 -> S^2, as follows: take a pair of regular values p and q and consider their preimages in S^3, which are oriented links P = f^-1(p) and Q = f^-1(q); then H(f) = the linking number of P and Q. Show that H(the Hopf map = ((z,w) -> z/w)) = 1. ======================================================================= 4/14 Poincare's Lemma implies H_c^n(R^n) = R. Later we'll see (by Poincare' duality) that the same is true for any connected n-manifold N in place of R^n. The isomorphism is by integration of a compactly supported n-form over N. Exercise: what if N is not connected? ----------------------------------------------------------------------- If f: M -> N is a proper smooth map between orientable manifolds of the same dimension, if N is connected, and if \alpha is a compactly supported n-form on N with integral 1, then deg(f) = \int_M f*\alpha. In other words, deg(f) is the induced homomorphism on cohomology H_c^n(N) = R -f*-> H_c^n(M) -> R composed with "summation" over the components of M. It follows that degree is a homotopy invariant: if f \~ g then deg(f) = deg(g). It also follows that deg(f g) = deg(f) deg(g). Try also to see these directly via the approach of (4/11). ----------------------------------------------------------------------- For each integer k there exists a map S^n -> S^n of degree k. (Hint: start with n = 1 and use cylidrical polar coordinates to "suspend".) ----------------------------------------------------------------------- For any M^n, there is a map M^n -> S^n of degree 1, and hence of arbitrary integer degree. When is there a such a map from M -> N? ----------------------------------------------------------------------- When is it true that deg(f) = deg(g) implies f \~ g? ======================================================================= 4/18 Recall that any connected manifold M has a 2-to-1 covering space M~ - the orientation double cover - which is disconnected iff M is orientable. We may regard M~ as a Z/2-bundle over M; in fact, a typical fiber is simply the unit vectors {+1,-1} in the line bundle given by the top exterior power of T*M. It follows that this line bundle is trivial - which is another way of saying M admits a non-vanishing n-form - iff M is orientable. If we take the tensor product of the line bundle with this Z/2 bundle (Z/2 acting on R by {+1,-1}), the resulting "twisted" line bundle is always trivial. In this way, a Z/2-orientation can always be represented by a nonvanising n-form, viewed as section of this twisted line bundle. ======================================================================= 4/21 The singular k-chains on a manifold M comprise the vector space C_k(M) of all (finite) R-linear combinations of smooth maps \sigma: \Delta_k -> M, where \Delta_k is the standard k-simplex. There is a natural boundary operator \del which is defined on each simplex and extended linearly. The image and kernel of \del, denoted B_k(M) and Z_k(M), are the boundaries and cycles. Check that \del \del = 0, and hence we get a complex C_*(M). Its homology, H_*(M) = Z_*(M)/B_*(M) is known as the singular homology of M. Show that H_0(M) = R^t, where t is the number of (path) compoments of M. Compare with the deRham cohomology H^n(M) - what's a nice basis for this? ----------------------------------------------------------------------- We can also define singular cochains C^k(M) by taking the dual space to the C_k(M). We get a (dual co)complex, with coboundary operator \delta, cocyles Z^k(M) and coboundaries B^k(M), whose (co)homology is singular cohmology H^*(M) = Z^*(M)/B^*(M). Verify that the singular cohomology groups of R^n are the same as the deRham cohomology groups of R^n. ----------------------------------------------------------------------- The singular (co)homology groups of a manifold also satisfy the Mayer-Vietoris theorem: given a covering of M by two open sets U and V, there is a long exact (co)homology sequence which computes H*(M) in terms of H*(U), H*(V) and H*(U\intV). It follows - from the above exercise, and the fact (we'll do this next week, when we discuss Riemanninan manifolds and the exponential map) that an n-manifold can be covered by charts which are diffeomorphic to R^n, and all of whose intersections are also diffeomorphic to R^n (or emepty) - that the deRham cohomology and singular cohomology of a manifold agree - this is the deRham Isomorphism Theorem. ----------------------------------------------------------------------- Integration gives a pairing between smooth k-forms and singular k-chains on M. Stokes Theorem extends linearly to this situation, and shows that the pairing descends to (co)homology. The deRham Isomorphism Theorem implies this pairing is nondegenerate. ======================================================================= 4/25 Let M -f-> R be a smooth function and suppose p is a critical point of f. Verify that Hess(f) is a well-defined symmetric bilinear form on the tangent space T_p M. By linear algebra (check this) we can thus define the rank, nullity, signature, index and co-index of Hess(f) at p. If the nullity is 0, we say p is nondegenerate. Prove the Morse Lemma: around a nondegenerate critical point p of index k we have f(x) = f(p) - x_1^2 - ... - x_k^2 + x_k+1^2 + ... + x_n^2 with respect to some coordinate system (centered at p). ----------------------------------------------------------------------- We say f is a Morse function if all its critical points are nondegenerate. In particular, a Morse function has isolated critical points. A good supply of Morse functions on M is obtained by embedding M in R^N, and taking (almost) any height function. [Hint: use Sard's theorem to check this!] ----------------------------------------------------------------------- Bott extended the definition - now called Morse-Bott - to include a function M -f-> R which is critical along a collection of embedded submanifolds M_1, ... , M_t with Hess(f) nondegenerate on the normal bundles to these M_j's. (The M_j's can have differing dimensions; when all are 0-dimensional, we recover the usual definition of Morse, since the normal bundle to {p} is simply T_p M.) For such an f, there is a Hopf-Morse index formula for the Euler number of M, but now the coefficients include the Euler numbers of the M_j's as well. Try to figure out this formula by considering examples of functions invariant under a Lie group action (for example, consider a surface of revolution and a height function along the axis). ======================================================================= 4/28 Describe handle decompositions of S^1, S^1 x S^1, a surface of genus g, a sphere S^n and a general product of spheres with respect to a Morse function on each (you might wish to pick an embedding of these manifolds into R^N and take a generic height function). ----------------------------------------------------------------------- Consider the function |z_0|^2 + 2|z_1|^2 + ... + 2^n|z_n|^2 on C^n+1. It restricts to an S^1-invariant function on S^2n-1, and thus a function CP^n -f-> R. Show that f is Morse with critical points of indices 0, 2, ... 2n. Describe the resulting handle decomposition. Try the analogous problem for RP^n, first finding a Morse function invariant under the antipodal map of S^n. ----------------------------------------------------------------------- Let M a compact n-manifold which carries a Morse function with exactly two critical points. Prove that M is homeomorphic to S^n. (It need not be diffeomorphic once n > 6....) ======================================================================= 5/2 We defined the Lefschetz number of a map M -f-> M and related it to the number of fixed points counted algebraically with a local degree of f. We also saw several corollaries, including the Hopf-Morse index theorems. Prove the Lefschetz formula. [Hint: consider the (transverse) intersection of graph(f) with the diagonal in M x M; see Guillemin-Pollack for details in case this intersection is transverse, in this case, all the Lefschetz degrees at the fixed points are +/-1, depending on the sign of det(df-I).] ----------------------------------------------------------------------- Give examples of vector fields on R^2 (and thus, R^n) with zeros of arbitrary Hopf index. [Hint: R^2 = C; see Milnor, Topology from the Differentiable Viewpoint for some pictures of examples.] ======================================================================= 5/5 We defined the stable and unstable manifolds S_p and U_p associated to each critical point p of a Morse function f on a Riemannian manifold M. These depend on the metric on M, but there is chain complex (the Morse-Smale-Floer-Witten-... Complex) whose homology is independent of the metric (and of f): let C_k(M,f) = R[{p \in M | index(f,p) = k}] (k = 0, 1, ... , n) be the R vector space with basis the critical points of Morse index k. These give a chain complex with boundary operators \del_k: C_k -> C_k-1 defined as follows (assume M is oriented): 1) Pick orientaions on S_p and U_p so that the tangent spaces T_pS_p+T_pU_p are oriented as M is. (If M is not orientable, forget signs and just consider Z/2 coefficients.) 2) For q \in C_k, let \del q = \Sum p where p \in C_k-1 and counts the number of gradient flow lines from p to q with a sign, depending on whether the orientation of U_q \int S_p agrees with the direction of flow. ----------------------------------------------------------------------- Work out the examples torus S^1 x S^1 with 4 critical points, and the "toothy" S^2 with 6 critical points examples with various metrics changing the flow lines. ----------------------------------------------------------------------- Now try the general case (you may want to forget signs, and do just a Z/2 version for a start): Show that \del_k-1 \del_k = 0, and thus there is a well defined homology; show that this homology is independent of the metric and the choice of f (in fact, it computes the singular homology of M, by the Mayer-Vietoris principle). ======================================================================= 5/9 Fill in the details of the Gauss-Bonnet-Lipschitz-Killing-Allendoerfer- Weil-Chern-Lashof theorem for an embedded hypersurface M^n -e-> R^n+1: deg(\nu:M -> S^n) = Euler(M)/2. You need to see that the Gauss (or rather, the Lipschitz-Killing) curvature det(d\nu) at p has sign which agrees with (-1)^k where k is the index of the critical point p for the height function f_v(p) = v.e(p) in the direction v, and use the change of variables formula to express deg(\nu) as (half) the average over v in S^n of the (signed) number of critical points for f_v (because, the critical points of f_v are the same as those for f_-v = -f_v, this double counts). ======================================================================= 5/12 The Morse inequalities are a consequence of the fact that the Morse-... complex computes the singular homology of M: i) c_k >= b_k for k = 0, 1, ..., n ii) c_k - c_k-1 + ... +/- c_0 >= b_k - b_k-1 + ... +/- b_0 where c_k = dim C_k and b_k = dim H_k = dim Z_k - dim B_k. Both are purely algebraic facts about a complex C_* and its homology H_*; in fact 1) is rather obvious since Z_k is a subspace of C_k. It's nice to think of these in terms of attaching handles to make M: i) there are at least as many k-handles as b_k ii) 1) the number of 1-handles is at least as many as b_1 plus the number of 1-handles needed to attach the 0-handles: c_1 >= b_1 + [c_0 - b_0] ii) 2) the number of 2-handles is at least as many as b_2 plus the number of 2-handles needed to attach the 1-handles: c_2 >= b_2 + [(c_1 - b_1) - (c_0 - b_0)] and so forth, noting that the preceding inequality implies [...] >= 0. ----------------------------------------------------------------------- The Morse inequalities are used both ways: to bound the topology of M, and prove the existence of critical points. For instance, a Morse function on RP^n must have n+1 critical points. (We noted in an earlier problem that a quadratic form on R^n+1 defines a function on RP^n, and it is Morse when it has distinct eigenvalues - the critical points correspond to the eigenlines.) ----------------------------------------------------------------------- How many critical points must a Morse function have on a surface of genus g? On the n-torus S^1 x ... S^1? On CP^n? ======================================================================= Have a great summer, year, and career! *"Copyleft" means that you have permission to use this material for non-commercial purposes as long as you acknowledge its source. Any use, in whole or in part, must include the "Copyleft 2000 by Rob Kusner" message or its equivalent. Commercial use without prior approval of the copyleft holder is strictly forbidden, and is punishable by the holder in any manner he sees fit!